S versus m Maple program and calculation results

 

# Finding the relationship between the m and the S of Lemma 3. 
# 
# 11 August 2007
# 
# So far, this program is only written for the case of odd b.
# 
> restart; b := 3;
                                      3
> with(numtheory);
[GIgcd, bigomega, cfrac, cfracpol, cyclotomic, divisors, factorEQ, factorset, 

  fermat, imagunit, index, integral_basis, invcfrac, invphi, issqrfree, 

  jacobi, kronecker, lambda, legendre, mcombine, mersenne, migcdex, minkowski, 

  mipolys, mlog, mobius, mroot, msqrt, nearestp, nthconver, nthdenom, 

  nthnumer, nthpow, order, pdexpand, phi, pi, pprimroot, primroot, quadres, 

  rootsunity, safeprime, sigma, sq2factor, sum2sqr, tau, thue]
> getexponent:= proc(c,p)
> local i,k;
> k:= c; i:= 0;
> if k mod p = 0 then 
> i:= 1;
> while k mod p^(i+1) = 0 do
> i:= i+1; 
> od;
> fi;
> i;
> end:
> getexponent(5400, 5);
                                      2
> getexponent(5400, 3);
                                      3
> twopart:= proc(n)
> local i,k;
> k:= n; i := 1;
> while k mod 2 = 0 do
> k := k/2;
> i := i*2;
> od;
> i;
> end:
> twopart(24);
                                      8
> twopart(124);
                                      4
# We first concentrate on finding S.  
# 
# The next preprocesses the g exponent for the small primes, up to 47, storing the answer in the incomplete array g_exp_value.  Note of course that the primes must be relatively prime to b.  Note we must do the 2 case separately. 
# 
> for i from 1 to 47 do g_exp_value[i] := 0; end do:
> for i from 2 to 15 do 
> p := ithprime(i); 
> if b mod p <> 0 then
> n := order(b, p); 
> testn := n/twopart(n); 
> while testn < n and `mod`(`&^`(b, testn)+1, p) <> 0 do testn := 2*testn end do; 
> if testn < n then g := getexponent(`&^`(b, testn)+1, p) else g := getexponent(`&^`(b, n)-1, p) end if; 
> g_exp_value[p] := g; 
> fi;
> end do:
> g_exp_value[2] := max(getexponent(b+1, 2), getexponent(b-1, 2));
                                      2
> g_exp_value[11];
                                      2
> getgvalue := proc(p)  
> local g,n, testn;    
> global b;    
> if b mod p <> 0 then  
> if p < 48 then 
> g := g_exp_value[p];
> else
> n := order(b, p);   
> testn := n/twopart(n);   
> while testn < n and `mod`(`&^`(b, testn)+1, p) <> 0 do 
> testn := 2*testn end do;   
> if testn < n then g := getexponent(`&^`(b, testn)+1, p) 
> else g := getexponent(`&^`(b, n)-1, p) end if;    
> fi;
> fi;  
> g;  
> end;
proc(p)  ...  end;
> findS:= proc(a)  
> local p,g, plist,i, S;  
> global b;  
> S:= 0;    
> plist:= factorset(a);    
> for i from 1 to nops(plist) do  
> p:= plist[i];  
> g:= getgvalue(p);
> S:= S+g*log(p)/log(a);
> od;
> S;
> end;
proc(a)  ...  end;
> findS(11);
                                      2
> for a from 2 to 25 do if igcd(a, b) = 1 then print(a, findS(a)) end if end do;
                                    2, 2
                                    4, 1
                                    5, 1
                                    7, 1
                                       2
                                    8, -
                                       3
                                2 ln(2)   ln(5) 
                            10, ------- + ------
                                ln(10)    ln(10)
                                    11, 2
                                    13, 1
                                2 ln(2)   ln(7) 
                            14, ------- + ------
                                ln(14)    ln(14)
                                        1
                                    16, -
                                        2
                                    17, 1
                                    19, 1
                                2 ln(2)   ln(5) 
                            20, ------- + ------
                                ln(20)    ln(20)
                               2 ln(2)   2 ln(11)
                           22, ------- + --------
                               ln(22)     ln(22) 
                                    23, 1
                                        1
                                    25, -
                                        2
# Now we find the m.   We find the order for each prime dividing a then take m to be the highest power of each prime in the order.  
> oddprimesm:= proc(a,n)  
> local alpha, gamma, p, oddprlist, m,i;  
> global b;  
> m:= 0;  
> oddprlist:= factorset(a) minus {2};  
> for i from 1 to nops(oddprlist) do  
> p:= oddprlist[i];  
> alpha := getexponent(a,p);  
> gamma := getexponent(b&^n-1,p);
> m := max(m, ceil(gamma/alpha));
> end do;
> m;
> end;
proc(a, n)  ...  end;
> oddprimesm(4561*(13*(8*11)*13), 15);
                                      2
# The idea of the next part is as follows:
# The odd factors of a give no problem.  So we need to see how to get the correct number of 2 factors.   
# 
# We have three cases: a odd, a mod 4 = 0 and a mod 4  = 2.
# 
# If a is odd, just treat the prime 2 as the odd primes.   
# 
# If a mod 4 = 0, then if n mod 4 = 2 then we need to consider the b^(n/2) + 1 mod a = 0 possibility.
# 
# If a mod 4 = 2 then we consider two cases:  
# (1.)  b^(n/2) + 1 mod a = 0 then we need to consider two cases: b^(n/2) + 1 mod 4 = 0 and b^(n/2) + 1 mod 4 = 2.  In this last case, we calculate g+h_0 + 1 and use this if it exceeds the oddprimesm value.  
# (2.)  b^(n/2) + 1 mod a = 2  We need to see if g+1 (note h_0 = 0) is bigger than the oddprimesm.
# 
> findm:= proc(a) 
> local n, m, alpha, gamma, gamma1, m2, g, h0;  
> global b; 
> if igcd(a,b) > 1 or iperfpow(a) <> FAIL then m := FAIL; 
> else 
> n:= order(b,a);
> m:= oddprimesm(a,n);
> alpha := getexponent(a,2);
> gamma:= getexponent(b&^n-1,2);
> 
> #Case 1 is when a is odd in which case m is oddprimesm.
> 
> #Case 2
> if a mod 4 = 0 then
> if n mod 4 = 2 and (b&^(n/2)+1) mod a = 0 then 
> gamma1 := getexponent(b&^(n/2)+1,2);
> m := max(m, ceil(gamma1/alpha));
> else
> m2:= ceil(gamma/alpha);
> m:= max(m, m2);
> end if;
> end if;
> # Case 3
> if a mod 4 = 2 then
> #Subcase 3.a
> if b&^n - 1 mod 4 = 2 then
> g := max( getexponent(b+1,2), getexponent(b-1,2));
> if g+1>m then m:= g+1; end if;
> end if;
> #note that in this case, m2 = 1, so m would be the oddprimesm.
> #Subcase 3.b
> if b&^n - 1 mod 4 = 0 then
> if n mod 2 = 0 and (b&^(n/2)+1) mod a = 0 then
> #two subsubcases
> if (b&^(n/2)+1) mod 4 = 0 then
> gamma1 := getexponent(b&^(n/2)+1,2);
> m := max(m, ceil(gamma1/alpha));
> end if;
> if (b&^(n/2)+1) mod 4 = 2 then
> g := max( getexponent(b+1,2), getexponent(b-1,2));
> h0 := getexponent(n,2);
> if (g+h0+1) > m then m:= g+h0+1 
> else
> m2:= ceil(gamma/alpha);
> m:= max(m,m2); 
> end if; 
> end if;
> else
> m2:= ceil(gamma/alpha);
> m := max(m, m2);
> end if;
> end if;
> end if;
> 
> end if;
> m; 
> end;
> 
proc(a)  ...  end;
> findS(2570); evalf(%); findm(2570);
                       2 ln(2)     ln(5)     ln(257) 
                       -------- + -------- + --------
                       ln(2570)   ln(2570)   ln(2570)
                                 1.088280322
                                     10
> 
# Now we compare the m and the S.  
# 
> c := 0; for a from 5 to 3000 do if `and`(igcd(a, b) = 1, iperfpow(a) = FAIL) then S := evalf(findS(a)); m := findm(a); if m-S > 1.9 then print(a, ifactor(a), m, S, m-S); c := c+1 end if end if end do; c;
                                      0
                  10, (2) (5), 5, 1.301029996, 3.698970004
                  34, (2) (17), 7, 1.196561632, 5.803438368
                            2                              
                 50, (2) (5) , 5, 0.7657757303, 4.234224270
                  58, (2) (29), 5, 1.170707280, 3.829292720
                        2                                  
                 68, (2)  (17), 3, 1.000000000, 2.000000000
                70, (2) (5) (7), 4, 1.163151220, 2.836848780
                  82, (2) (41), 6, 1.157293248, 4.842706752
                 106, (2) (53), 5, 1.148634338, 3.851365662
               110, (2) (5) (11), 4, 1.657601396, 2.342398604
               130, (2) (5) (13), 4, 1.142402111, 2.857597889
                 146, (2) (73), 5, 1.139085452, 3.860914548
                         2                                  
                 164, (2)  (41), 3, 1.000000000, 2.000000000
               170, (2) (5) (17), 6, 1.134963860, 4.865036140
                 178, (2) (89), 6, 1.133766140, 4.866233860
               190, (2) (5) (19), 4, 1.132102916, 2.867897084
                 194, (2) (97), 7, 1.131580457, 5.868419543
                 202, (2) (101), 5, 1.130578792, 3.869421208
                 226, (2) (113), 7, 1.127874312, 5.872125688
               230, (2) (5) (23), 4, 1.127461764, 2.872538236
               238, (2) (7) (17), 6, 1.126665368, 4.873334632
                             3                              
                 250, (2) (5) , 5, 0.5425615281, 4.457438472
                 274, (2) (137), 6, 1.123486792, 4.876513208
               290, (2) (5) (29), 5, 1.122250747, 3.877749253
                 298, (2) (149), 5, 1.121666808, 3.878333192
               310, (2) (5) (31), 4, 1.120829503, 2.879170497
                             2                              
                338, (2) (13) , 3, 0.6785527350, 2.321447265
                      2                                       
              340, (2)  (5) (17), 3, 0.9999999999, 2.000000000
                 346, (2) (173), 5, 1.118558871, 3.881441129
                           2                                  
               350, (2) (5)  (7), 4, 0.8435812244, 3.156418776
                         2                                  
                 356, (2)  (89), 3, 1.000000000, 2.000000000
               370, (2) (5) (37), 4, 1.117214311, 2.882785689
               374, (2) (11) (17), 6, 1.521760464, 4.478239536
                 386, (2) (193), 7, 1.116381146, 5.883618854
                         2                                  
                 388, (2)  (97), 3, 1.000000000, 2.000000000
                 394, (2) (197), 5, 1.115981674, 3.884018326
               406, (2) (7) (29), 4, 1.115402335, 2.884597665
               410, (2) (5) (41), 5, 1.115214274, 3.884785726
               430, (2) (5) (43), 4, 1.114309323, 2.885690677
               442, (2) (13) (17), 6, 1.113792796, 4.886207204
                        2                                    
                452, (2)  (113), 3, 0.9999999999, 2.000000000
                 466, (2) (233), 6, 1.112813516, 4.887186484
               470, (2) (5) (47), 4, 1.112656800, 2.887343200
                       2                                      
               476, (2)  (7) (17), 3, 1.000000000, 2.000000000
                 482, (2) (241), 6, 1.112197062, 4.887802938
                               2                              
               490, (2) (5) (7) , 4, 0.7977589631, 3.202241037
                 505, (5) (101), 3, 1.000000000, 2.000000000
                514, (2) (257), 11, 1.111041716, 9.888958284
               530, (2) (5) (53), 5, 1.110499087, 3.889500913
                 538, (2) (269), 5, 1.110235810, 3.889764190
                        2                                   
                548, (2)  (137), 3, 1.000000000, 2.000000000
                           2                                  
               550, (2) (5)  (11), 4, 1.234805662, 2.765194338
                 562, (2) (281), 6, 1.109475950, 4.890524050
               574, (2) (7) (41), 5, 1.109111856, 3.890888144
                             2                              
                578, (2) (17) , 7, 0.6634890627, 6.336510937
                 586, (2) (293), 5, 1.108757634, 3.891242366
               590, (2) (5) (59), 4, 1.108641672, 2.891358328
               610, (2) (5) (61), 4, 1.108076965, 2.891923035
                 634, (2) (317), 5, 1.107430552, 3.892569448
               638, (2) (11) (29), 4, 1.478612658, 2.521387342
               646, (2) (17) (19), 6, 1.107119248, 4.892880752
                          2                                   
              650, (2) (5)  (13), 4, 0.8585310108, 3.141468989
               670, (2) (5) (67), 4, 1.106518764, 2.893481236
                 674, (2) (337), 6, 1.106421418, 4.893578582
                 706, (2) (353), 8, 1.105668878, 6.894331122
               710, (2) (5) (71), 4, 1.105577944, 2.894422056
               730, (2) (5) (73), 5, 1.105133096, 3.894866904
               742, (2) (7) (53), 4, 1.104873741, 2.895126259
               754, (2) (13) (29), 4, 1.104619793, 2.895380207
                755, (5) (151), 3, 0.9999999999, 2.000000000
             770, (2) (5) (7) (11), 4, 1.465070845, 2.534929155
                        2                                   
                772, (2)  (193), 3, 1.000000000, 2.000000000
                 778, (2) (389), 5, 1.104127333, 3.895872667
               782, (2) (17) (23), 6, 1.104047177, 4.895952823
               790, (2) (5) (79), 4, 1.103888453, 2.896111547
                 802, (2) (401), 7, 1.103654243, 5.896345757
                 818, (2) (409), 5, 1.103348950, 3.896651050
                       2                                      
               820, (2)  (5) (41), 3, 1.000000000, 2.000000000
               830, (2) (5) (83), 4, 1.103125023, 2.896874977
                          2                                   
              850, (2) (5)  (17), 6, 0.8641573572, 5.135842643
                      2                                        
              884, (2)  (13) (17), 3, 0.9999999998, 2.000000000
               890, (2) (5) (89), 5, 1.102065171, 3.897934829
                 898, (2) (449), 9, 1.101930860, 7.898069140
               902, (2) (11) (41), 5, 1.454256806, 3.545743194
             910, (2) (5) (7) (13), 4, 1.101732269, 2.898267731
                 914, (2) (457), 5, 1.101666824, 3.898333176
                 922, (2) (461), 5, 1.101537038, 3.898462962
                        2                                   
                932, (2)  (233), 3, 1.000000000, 2.000000000
                          2                                   
              950, (2) (5)  (19), 4, 0.8663609987, 3.133639001
                        2                                    
                964, (2)  (241), 3, 0.9999999999, 2.000000000
               970, (2) (5) (97), 6, 1.100787748, 4.899212252
               986, (2) (17) (29), 6, 1.100548554, 4.899451446
              1010, (2) (5) (101), 5, 1.100198999, 3.899801001
                1018, (2) (509), 5, 1.100084854, 3.899915146
               1022, (2) (7) (73), 4, 1.100028213, 2.899971787
                         2                                   
                1028, (2)  (257), 5, 1.000000000, 4.000000000
              1030, (2) (5) (103), 4, 1.099915785, 2.900084215
                1042, (2) (521), 6, 1.099749235, 4.900250765
              1054, (2) (17) (31), 6, 1.099585137, 4.900414863
                              2                              
                1058, (2) (23) , 3, 0.6492964568, 2.350703543
              1066, (2) (13) (41), 5, 1.099423426, 3.900576574
              1070, (2) (5) (107), 4, 1.099370043, 2.900629957
                1082, (2) (541), 3, 1.099211420, 1.900788580
              1090, (2) (5) (109), 4, 1.099106924, 2.900893076
              1102, (2) (19) (29), 4, 1.098952014, 2.901047986
                1114, (2) (557), 5, 1.098799258, 3.901200742
              1118, (2) (13) (43), 3, 1.098748808, 1.901251192
                         2                                   
                1124, (2)  (281), 3, 1.000000000, 2.000000000
                1126, (2) (563), 3, 1.098648602, 1.901351398
              1130, (2) (5) (113), 6, 1.098598841, 4.901401159
                1138, (2) (569), 6, 1.098499994, 4.901500006
                       2                                      
              1148, (2)  (7) (41), 3, 1.000000000, 2.000000000
                           2                                   
              1150, (2) (5)  (23), 4, 0.8699838960, 3.130016104
                1154, (2) (577), 7, 1.098304952, 5.901695048
               1162, (2) (7) (83), 3, 1.098208728, 1.901791272
              1166, (2) (11) (53), 4, 1.437741975, 2.562258025
                1174, (2) (587), 3, 1.098065975, 1.901934025
                1186, (2) (593), 7, 1.097925082, 5.902074918
             1190, (2) (5) (7) (17), 6, 1.097878523, 4.902121477
                1198, (2) (599), 3, 1.097786006, 1.902213994
                1202, (2) (601), 3, 1.097740043, 1.902259957
                                2                             
              1210, (2) (5) (11) , 4, 1.097648704, 2.902351296
              1222, (2) (13) (47), 3, 1.097513136, 1.902486864
                1234, (2) (617), 6, 1.097379264, 4.902620736
               1246, (2) (7) (89), 5, 1.097247049, 3.902752951
                             4                              
                1250, (2) (5) , 5, 0.4201058443, 4.579894156
                1255, (5) (251), 4, 0.9999999999, 3.000000000
              1258, (2) (17) (37), 6, 1.097116455, 4.902883545
              1270, (2) (5) (127), 4, 1.096987446, 2.903012554
                           2                                   
              1274, (2) (7)  (13), 3, 0.8247863580, 2.175213642
                1282, (2) (641), 10, 1.096859988, 8.903140012
                       2                                       
              1292, (2)  (17) (19), 3, 1.000000000, 2.000000000
                1294, (2) (647), 3, 1.096734047, 1.903265953
                1306, (2) (653), 5, 1.096609591, 3.903390409
              1310, (2) (5) (131), 4, 1.096568430, 2.903431570
                1318, (2) (659), 3, 1.096486589, 1.903513411
             1330, (2) (5) (7) (19), 4, 1.096365010, 2.903634990
              1334, (2) (23) (29), 4, 1.096324795, 2.903675205
                1346, (2) (673), 6, 1.096205069, 4.903794931
                         2                                   
                1348, (2)  (337), 3, 1.000000000, 2.000000000
                1354, (2) (677), 5, 1.096126006, 3.903873994
               1358, (2) (7) (97), 6, 1.096086699, 4.903913301
                1366, (2) (683), 3, 1.096008525, 1.903991475
              1370, (2) (5) (137), 5, 1.095969656, 3.904030344
              1378, (2) (13) (53), 4, 1.095892353, 2.904107647
                1379, (7) (197), 3, 1.000000000, 2.000000000
              1390, (2) (5) (139), 4, 1.095777467, 2.904222533
              1394, (2) (17) (41), 6, 1.095739452, 4.904260548
                1402, (2) (701), 5, 1.095663839, 3.904336161
                         2                                   
                1412, (2)  (353), 4, 1.000000000, 3.000000000
              1414, (2) (7) (101), 4, 1.095551446, 2.904448554
                1418, (2) (709), 3, 1.095514251, 1.904485749
              1426, (2) (23) (31), 3, 1.095440262, 1.904559738
            1430, (2) (5) (11) (13), 4, 1.425445234, 2.574554766
                1438, (2) (719), 3, 1.095330266, 1.904669734
                           2                                   
              1450, (2) (5)  (29), 5, 0.8741241107, 4.125875889
              1462, (2) (17) (43), 6, 1.095113745, 4.904886255
                1466, (2) (733), 3, 1.095078098, 1.904921902
                1486, (2) (743), 3, 1.094901706, 1.905098294
              1490, (2) (5) (149), 5, 1.094866790, 3.905133210
              1498, (2) (7) (107), 3, 1.094797316, 1.905202684
              1510, (2) (5) (151), 4, 1.094693985, 2.905306015
                1514, (2) (757), 3, 1.094659774, 1.905340226
                1522, (2) (761), 6, 1.094591695, 4.905408305
              1526, (2) (7) (109), 3, 1.094557826, 1.905442174
              1534, (2) (13) (59), 3, 1.094490426, 1.905509574
                1538, (2) (769), 7, 1.094456894, 5.905543106
                1546, (2) (773), 5, 1.094390161, 3.905609839
                           2                                   
              1550, (2) (5)  (31), 4, 0.8752668854, 3.124733115
              1558, (2) (19) (41), 5, 1.094290880, 3.905709120
                       2                                       
              1564, (2)  (17) (23), 3, 1.000000000, 2.000000000
              1570, (2) (5) (157), 4, 1.094192568, 2.905807432
              1582, (2) (7) (113), 6, 1.094095207, 4.905904793
              1586, (2) (13) (61), 3, 1.094062962, 1.905937038
                1594, (2) (797), 5, 1.093998780, 3.906001220
              1598, (2) (17) (47), 6, 1.093966843, 4.906033157
                         2                                   
                1604, (2)  (401), 3, 1.000000000, 2.000000000
              1606, (2) (11) (73), 4, 1.418755220, 2.581244780
             1610, (2) (5) (7) (23), 4, 1.093871637, 2.906128363
                1618, (2) (809), 6, 1.093808666, 4.906191334
              1630, (2) (5) (163), 4, 1.093714948, 2.906285052
                1642, (2) (821), 5, 1.093622103, 3.906377897
                1654, (2) (827), 3, 1.093530115, 1.906469885
                1658, (2) (829), 3, 1.093499640, 1.906500360
                           2                                   
              1666, (2) (7)  (17), 6, 0.8311226163, 5.168877384
              1670, (2) (5) (167), 4, 1.093408774, 2.906591226
                1678, (2) (839), 3, 1.093348656, 1.906651344
                              2                              
                1682, (2) (29) , 5, 0.6399781001, 4.360021900
                                2                              
              1690, (2) (5) (13) , 4, 0.7481592660, 3.251840734
                                2                             
              1694, (2) (7) (11) , 3, 1.093229504, 1.906770496
                      2    2                                   
             1700, (2)  (5)  (17), 3, 0.7836306899, 2.216369310
              1702, (2) (23) (37), 3, 1.093170462, 1.906829538
                1714, (2) (857), 6, 1.093082557, 4.906917443
                1726, (2) (863), 3, 1.092995428, 1.907004572
              1730, (2) (5) (173), 5, 1.092966556, 3.907033444
              1742, (2) (13) (67), 3, 1.092880445, 1.907119555
                           3                                  
              1750, (2) (5)  (7), 4, 0.6617646797, 3.338235320
                1762, (2) (881), 7, 1.092738586, 5.907261414
                1774, (2) (887), 3, 1.092654446, 1.907345554
                       2                                      
              1780, (2)  (5) (89), 3, 1.000000000, 2.000000000
              1786, (2) (19) (47), 3, 1.092571024, 1.907428976
              1790, (2) (5) (179), 4, 1.092543374, 2.907456626
                         2                                   
                1796, (2)  (449), 4, 1.000000000, 3.000000000
              1798, (2) (29) (31), 4, 1.092488309, 2.907511691
              1802, (2) (17) (53), 6, 1.092460893, 4.907539107
              1810, (2) (5) (181), 4, 1.092406291, 2.907593709
                1822, (2) (911), 3, 1.092324959, 1.907675041
              1834, (2) (7) (131), 3, 1.092244303, 1.907755697
              1846, (2) (13) (71), 3, 1.092164312, 1.907835688
                           2                                   
              1850, (2) (5)  (37), 4, 0.8782004615, 3.121799538
                1858, (2) (929), 8, 1.092084976, 6.907915024
            1870, (2) (5) (11) (17), 6, 1.410295742, 4.589704258
                1882, (2) (941), 5, 1.091928233, 3.908071767
              1886, (2) (23) (41), 5, 1.091902355, 3.908097645
                1894, (2) (947), 3, 1.091850807, 1.908149193
              1898, (2) (13) (73), 4, 1.091825136, 2.908174864
                1906, (2) (953), 6, 1.091773999, 4.908226001
              1910, (2) (5) (191), 4, 1.091748532, 2.908251468
              1918, (2) (7) (137), 5, 1.091697800, 3.908302200
              1930, (2) (5) (193), 6, 1.091622202, 4.908377798
                       2                                      
              1940, (2)  (5) (97), 3, 1.000000000, 2.000000000
                1942, (2) (971), 3, 1.091547196, 1.908452804
                1954, (2) (977), 7, 1.091472773, 5.908527227
              1958, (2) (11) (89), 5, 1.407806521, 3.592193479
                1966, (2) (983), 3, 1.091398926, 1.908601074
              1970, (2) (5) (197), 5, 1.091374437, 3.908625563
                       2                                       
              1972, (2)  (17) (29), 3, 1.000000000, 2.000000000
              1978, (2) (23) (43), 3, 1.091325646, 1.908674354
              1990, (2) (5) (199), 4, 1.091252926, 2.908747074
                2005, (5) (401), 3, 0.9999999999, 2.000000000
              2006, (2) (17) (59), 6, 1.091156823, 4.908843177
              2014, (2) (19) (53), 4, 1.091109134, 2.908890866
                2018, (2) (1009), 6, 1.091085379, 4.908914621
                       2                                       
              2020, (2)  (5) (101), 3, 1.000000000, 2.000000000
                2026, (2) (1013), 5, 1.091038047, 3.908961953
             2030, (2) (5) (7) (29), 4, 1.091014469, 2.908985531
                2038, (2) (1019), 3, 1.090967489, 1.909032511
                           2                                   
              2050, (2) (5)  (41), 5, 0.8798401012, 4.120159899
              2054, (2) (13) (79), 3, 1.090874224, 1.909125776
                        3                                    
               2056, (2)  (257), 4, 0.9091373692, 3.090862631
                2062, (2) (1031), 3, 1.090827935, 1.909172065
                2066, (2) (1033), 5, 1.090804876, 3.909195124
              2074, (2) (17) (61), 6, 1.090758925, 4.909241075
                         2                                   
                2084, (2)  (521), 3, 1.000000000, 2.000000000
              2086, (2) (7) (149), 4, 1.090690416, 2.909309584
            2090, (2) (5) (11) (19), 4, 1.404326366, 2.595673634
                2098, (2) (1049), 6, 1.090622403, 4.909377597
                       2                                       
              2108, (2)  (17) (31), 3, 1.000000000, 2.000000000
              2110, (2) (5) (211), 4, 1.090554879, 2.909445121
                2122, (2) (1061), 5, 1.090487838, 3.909512162
                       2                                       
              2132, (2)  (13) (41), 3, 1.000000000, 2.000000000
              2134, (2) (11) (97), 6, 1.403227482, 4.596772518
              2146, (2) (29) (37), 4, 1.090355178, 2.909644822
                           2                                   
              2150, (2) (5)  (43), 4, 0.8805859392, 3.119414061
              2158, (2) (13) (83), 3, 1.090289547, 1.909710453
              2162, (2) (23) (47), 3, 1.090267773, 1.909732227
             2170, (2) (5) (7) (31), 4, 1.090224375, 2.909775625
                2182, (2) (1091), 3, 1.090159656, 1.909840344
                2186, (2) (1093), 3, 1.090138183, 1.909861817
                2194, (2) (1097), 6, 1.090095384, 4.909904616
                2206, (2) (1103), 3, 1.090031553, 1.909968447
            2210, (2) (5) (13) (17), 6, 1.090010373, 4.909989627
                2218, (2) (1109), 5, 1.089968158, 3.910031842
              2222, (2) (11) (101), 4, 1.401113040, 2.598886960
              2230, (2) (5) (223), 4, 1.089905193, 2.910094807
                2234, (2) (1117), 3, 1.089884300, 1.910115700
              2242, (2) (19) (59), 3, 1.089842654, 1.910157346
                           2                                   
              2254, (2) (7)  (23), 3, 0.8377347083, 2.162265292
                2258, (2) (1129), 5, 1.089759921, 3.910240079
                       2                                       
              2260, (2)  (5) (113), 3, 1.000000000, 2.000000000
              2270, (2) (5) (227), 4, 1.089698354, 2.910301646
                         2                                   
                2276, (2)  (569), 3, 1.000000000, 2.000000000
              2278, (2) (17) (67), 6, 1.089657537, 4.910342463
              2290, (2) (5) (229), 4, 1.089596648, 2.910403352
                2302, (2) (1151), 3, 1.089536159, 1.910463841
                2306, (2) (1153), 9, 1.089516084, 7.910483916
                         2                                   
                2308, (2)  (577), 3, 1.000000000, 2.000000000
              2314, (2) (13) (89), 5, 1.089476065, 3.910523935
                2326, (2) (1163), 3, 1.089416363, 1.910583637
              2330, (2) (5) (233), 5, 1.089396548, 3.910603452
              2338, (2) (7) (167), 3, 1.089357047, 1.910642953
                           2                                   
              2350, (2) (5)  (47), 4, 0.8819543169, 3.118045683
                2362, (2) (1181), 5, 1.089239555, 3.910760445
                                2                              
              2366, (2) (7) (13) , 3, 0.7590664477, 2.240933552
                         2                                   
                2372, (2)  (593), 3, 1.000000000, 2.000000000
                2374, (2) (1187), 3, 1.089181370, 1.910818630
              2378, (2) (29) (41), 5, 1.089162058, 3.910837942
                     2                                           
            2380, (2)  (5) (7) (17), 3, 0.9999999999, 2.000000000
                2386, (2) (1193), 6, 1.089123554, 4.910876446
              2390, (2) (5) (239), 4, 1.089104364, 2.910895636
                2402, (2) (1201), 5, 1.089047033, 3.910952967
              2410, (2) (5) (241), 5, 1.089009012, 3.910990988
              2414, (2) (17) (71), 6, 1.088990061, 4.911009939
              2422, (2) (7) (173), 4, 1.088952277, 2.911047723
                2426, (2) (1213), 3, 1.088933444, 1.911066556
                2434, (2) (1217), 9, 1.088895894, 7.911104106
              2438, (2) (23) (53), 4, 1.088877178, 2.911122822
                2446, (2) (1223), 3, 1.088839860, 1.911160140
                           2    2                              
              2450, (2) (5)  (7) , 4, 0.6332318871, 3.366768113
                2458, (2) (1229), 5, 1.088784169, 3.911215831
                         2                                   
                2468, (2)  (617), 3, 1.000000000, 2.000000000
            2470, (2) (5) (13) (19), 4, 1.088728819, 2.911271181
                2474, (2) (1237), 3, 1.088710444, 1.911289556
              2482, (2) (17) (73), 6, 1.088673806, 4.911326194
              2486, (2) (11) (113), 6, 1.395353332, 4.604646668
                       2                                      
              2492, (2)  (7) (89), 3, 1.000000000, 2.000000000
              2494, (2) (29) (43), 4, 1.088619126, 2.911380874
                2498, (2) (1249), 7, 1.088600973, 5.911399027
              2506, (2) (7) (179), 3, 1.088564776, 1.911435224
              2510, (2) (5) (251), 4, 1.088546731, 2.911453269
                       2                                       
              2516, (2)  (17) (37), 3, 1.000000000, 2.000000000
                2518, (2) (1259), 3, 1.088510751, 1.911489249
              2522, (2) (13) (97), 6, 1.088492814, 4.911507186
            2530, (2) (5) (11) (23), 4, 1.394468157, 2.605531843
              2534, (2) (7) (181), 3, 1.088439218, 1.911560782
              2542, (2) (31) (41), 5, 1.088403664, 3.911596336
                2554, (2) (1277), 5, 1.088350596, 3.911649404
                         2                                   
                2564, (2)  (641), 5, 1.000000000, 4.000000000
                2566, (2) (1283), 3, 1.088297839, 1.911702161
              2570, (2) (5) (257), 10, 1.088280322, 8.911719678
                2578, (2) (1289), 6, 1.088245391, 4.911754609
             2590, (2) (5) (7) (37), 4, 1.088193249, 2.911806751
                2602, (2) (1301), 5, 1.088141409, 3.911858591
                2614, (2) (1307), 3, 1.088089868, 1.911910132
            2618, (2) (7) (11) (17), 6, 1.392754419, 4.607245581
              2626, (2) (13) (101), 4, 1.088038622, 2.911961378
              2630, (2) (5) (263), 4, 1.088021606, 2.911978394
                2638, (2) (1319), 3, 1.087987670, 1.912012330
                2642, (2) (1321), 3, 1.087970750, 1.912029250
                           2                                   
              2650, (2) (5)  (53), 5, 0.8837536002, 4.116246400
                              3                              
                2662, (2) (11) , 3, 0.7838488412, 2.216151159
              2674, (2) (7) (191), 3, 1.087836539, 1.912163461
              2678, (2) (13) (103), 3, 1.087819904, 1.912180096
              2686, (2) (17) (79), 6, 1.087786728, 4.912213272
              2690, (2) (5) (269), 5, 1.087770186, 3.912229814
                         2                                   
                2692, (2)  (673), 3, 1.000000000, 2.000000000
              2698, (2) (19) (71), 3, 1.087737195, 1.912262805
              2702, (2) (7) (193), 6, 1.087720745, 4.912279255
              2710, (2) (5) (271), 4, 1.087687937, 2.912312063
              2714, (2) (23) (59), 3, 1.087671579, 1.912328421
                       2                                      
              2716, (2)  (7) (97), 3, 1.000000000, 2.000000000
                2722, (2) (1361), 7, 1.087638952, 5.912361048
              2726, (2) (29) (47), 4, 1.087622684, 2.912377316
                2734, (2) (1367), 3, 1.087590237, 1.912409763
                       2                                       
              2740, (2)  (5) (137), 3, 1.000000000, 2.000000000
                2746, (2) (1373), 5, 1.087541789, 3.912458211
                           3                                   
              2750, (2) (5)  (11), 4, 0.9838581106, 3.016141889
              2758, (2) (7) (197), 4, 1.087493605, 2.912506395
                2762, (2) (1381), 3, 1.087477602, 1.912522398
              2770, (2) (5) (277), 4, 1.087445683, 2.912554317
              2774, (2) (19) (73), 4, 1.087429767, 2.912570233
              2782, (2) (13) (107), 3, 1.087398021, 1.912601979
                      2                                        
             2788, (2)  (17) (41), 3, 0.9999999999, 2.000000000
              2806, (2) (23) (61), 3, 1.087303464, 1.912696536
              2810, (2) (5) (281), 5, 1.087287802, 3.912712198
               2818, (2) (1409), 10, 1.087256564, 8.912743436
              2822, (2) (17) (83), 6, 1.087240986, 4.912759014
                        3                                    
               2824, (2)  (353), 3, 0.9127667925, 2.087233208
              2830, (2) (5) (283), 4, 1.087209913, 2.912790087
              2834, (2) (13) (109), 3, 1.087194418, 1.912805582
                           2                                   
              2842, (2) (7)  (29), 4, 0.8424646015, 3.157535398
                2854, (2) (1427), 3, 1.087117351, 1.912882649
                2866, (2) (1433), 6, 1.087071434, 4.912928566
             2870, (2) (5) (7) (41), 5, 1.087056182, 3.912943818
                2878, (2) (1439), 3, 1.087025757, 1.912974243
                                2                              
              2890, (2) (5) (17) , 6, 0.7314514995, 5.268548500
                2902, (2) (1451), 3, 1.086935115, 1.913064885
                2906, (2) (1453), 3, 1.086920099, 1.913079901
              2914, (2) (31) (47), 3, 1.086890144, 1.913109856
                       2                                       
              2924, (2)  (17) (43), 3, 1.000000000, 2.000000000
              2930, (2) (5) (293), 5, 1.086830542, 3.913169458
              2938, (2) (13) (113), 6, 1.086800894, 4.913199106
                           2                                   
              2950, (2) (5)  (59), 4, 0.8853139998, 3.114686000
                2962, (2) (1481), 6, 1.086712551, 4.913287449
                2974, (2) (1487), 3, 1.086668714, 1.913331286
                2978, (2) (1489), 6, 1.086654151, 4.913345849
                2986, (2) (1493), 5, 1.086625098, 3.913374902
            2990, (2) (5) (13) (23), 4, 1.086610608, 2.913389392
                2998, (2) (1499), 3, 1.086581701, 1.913418299
                                     378
> c := 0; for a from 20000 to 24000 do if `and`(igcd(a, b) = 1, iperfpow(a) = FAIL) then S := evalf(findS(a)); m := findm(a); if m-S > 1.9 then c := c+1; print(a, ifactor(a), m, S, m-S); findmdetails(a) end if end if end do; c;
                                      0
             20002, (2) (73) (137), 5, 1.069989504, 3.930010496
               20005, (5) (4001), 4, 1.000000000, 3.000000000
               20014, (2) (10007), 3, 1.069985266, 1.930014734
               20018, (2) (10009), 5, 1.069983854, 3.930016146
           20026, (2) (17) (19) (31), 6, 1.069981030, 4.930018970
             20030, (2) (5) (2003), 4, 1.069979619, 2.930020381
                         2                                    
               20036, (2)  (5009), 3, 1.000000000, 2.000000000
             20038, (2) (43) (233), 5, 1.069976798, 3.930023202
                           2                                    
             20050, (2) (5)  (401), 6, 0.9075012949, 5.092498705
                     2                                            
           20060, (2)  (5) (17) (59), 3, 0.9999999999, 2.000000000
             20062, (2) (7) (1433), 5, 1.069968343, 3.930031657
               20074, (2) (10037), 5, 1.069964120, 3.930035880
                       4                                        
             20080, (2)  (5) (251), 4, 0.8600759808, 3.139924019
                            2                                  
             20086, (2) (11)  (83), 3, 1.069959900, 1.930040100
                             2                                   
           20090, (2) (5) (7)  (41), 5, 0.8735601717, 4.126439828
             20098, (2) (13) (773), 4, 1.069955683, 2.930044317
                                 2                              
             20102, (2) (19) (23) , 3, 0.7535117677, 2.246488232
             20110, (2) (5) (2011), 4, 1.069951469, 2.930048531
             20114, (2) (89) (113), 6, 1.069950065, 4.930049935
               20122, (2) (10061), 5, 1.069947258, 3.930052742
             20126, (2) (29) (347), 4, 1.069945855, 2.930054145
               20134, (2) (10067), 3, 1.069943050, 1.930056950
               20138, (2) (10069), 3, 1.069941648, 1.930058352
             20146, (2) (7) (1439), 3, 1.069938845, 1.930061155
                         2                                        
           20150, (2) (5)  (13) (31), 4, 0.9075477277, 3.092452272
               20158, (2) (10079), 3, 1.069934643, 1.930065357
             20162, (2) (17) (593), 7, 1.069933243, 5.930066757
             20170, (2) (5) (2017), 6, 1.069930444, 4.930069556
                      2                                         
            20180, (2)  (5) (1009), 3, 0.9999999999, 2.000000000
               20182, (2) (10091), 3, 1.069926248, 1.930073752
               20186, (2) (10093), 3, 1.069924850, 1.930075150
             20194, (2) (23) (439), 3, 1.069922055, 1.930077945
               20206, (2) (10103), 3, 1.069917865, 1.930082135
            20210, (2) (5) (43) (47), 4, 1.069916470, 2.930083530
                                  2                              
           20230, (2) (5) (7) (17) , 6, 0.7841570333, 5.215842967
             20242, (2) (29) (349), 4, 1.069905314, 2.930094686
             20246, (2) (53) (191), 4, 1.069903921, 2.930096079
           20254, (2) (13) (19) (41), 5, 1.069901136, 3.930098864
               20266, (2) (10133), 5, 1.069896960, 3.930103040
             20270, (2) (5) (2027), 4, 1.069895570, 2.930104430
                       2                                        
             20276, (2)  (37) (137), 3, 1.000000000, 2.000000000
               20278, (2) (10139), 3, 1.069892788, 1.930107212
             20290, (2) (5) (2029), 4, 1.069888619, 2.930111381
             20294, (2) (73) (139), 4, 1.069887230, 2.930112770
               20302, (2) (10151), 3, 1.069884453, 1.930115547
             20314, (2) (7) (1451), 3, 1.069880290, 1.930119710
                         2                                    
               20324, (2)  (5081), 3, 1.000000000, 2.000000000
               20326, (2) (10163), 3, 1.069876130, 1.930123870
           20330, (2) (5) (19) (107), 4, 1.069874744, 2.930125256
                     2                                            
           20332, (2)  (13) (17) (23), 3, 1.000000000, 2.000000000
               20338, (2) (10169), 6, 1.069871973, 4.930128027
             20342, (2) (7) (1453), 3, 1.069870588, 1.930129412
                         2                                       
           20350, (2) (5)  (11) (37), 4, 1.149342708, 2.850657292
               20354, (2) (10177), 7, 1.069866434, 5.930133566
               20362, (2) (10181), 5, 1.069863667, 3.930136333
             20366, (2) (17) (599), 6, 1.069862284, 4.930137716
             20374, (2) (61) (167), 3, 1.069859518, 1.930140482
             20378, (2) (23) (443), 3, 1.069858136, 1.930141864
               20386, (2) (10193), 7, 1.069855373, 5.930144627
             20390, (2) (5) (2039), 4, 1.069853992, 2.930146008
            20398, (2) (7) (31) (47), 3, 1.069851230, 1.930148770
                               2                              
               20402, (2) (101) , 5, 0.6047747754, 4.395225225
           20410, (2) (5) (13) (157), 4, 1.069847091, 2.930152909
             20414, (2) (59) (173), 4, 1.069845712, 2.930154288
               20422, (2) (10211), 3, 1.069842954, 1.930157046
             20434, (2) (17) (601), 6, 1.069838820, 4.930161180
             20438, (2) (11) (929), 7, 1.311435302, 5.688564698
               20446, (2) (10223), 3, 1.069834690, 1.930165310
                           2                                    
             20450, (2) (5)  (409), 5, 0.9076853816, 4.092314618
             20458, (2) (53) (193), 6, 1.069830562, 4.930169438
             20462, (2) (13) (787), 3, 1.069829186, 1.930170814
                     2                                           
           20468, (2)  (7) (17) (43), 3, 1.000000000, 2.000000000
            20470, (2) (5) (23) (89), 5, 1.069826436, 3.930173564
             20474, (2) (29) (353), 7, 1.069825062, 5.930174938
               20494, (2) (10247), 3, 1.069818195, 1.930181805
             20498, (2) (37) (277), 3, 1.069816823, 1.930183177
                       2    3                                   
             20500, (2)  (5)  (41), 3, 0.6757839028, 2.324216097
               20506, (2) (10253), 5, 1.069814079, 3.930185921
            20510, (2) (5) (7) (293), 4, 1.069812707, 2.930187293
               20518, (2) (10259), 3, 1.069809965, 1.930190035
             20530, (2) (5) (2053), 4, 1.069805855, 2.930194145
               20542, (2) (10271), 3, 1.069801747, 1.930198253
               20546, (2) (10273), 7, 1.069800378, 5.930199622
             20554, (2) (43) (239), 3, 1.069797642, 1.930202358
             20558, (2) (19) (541), 3, 1.069796274, 1.930203726
                       4                                        
             20560, (2)  (5) (257), 3, 0.8604088186, 2.139591181
                       2                                       
             20564, (2)  (53) (97), 3, 1.000000000, 2.000000000
           20566, (2) (7) (13) (113), 6, 1.069793540, 4.930206460
                              2                                  
           20570, (2) (5) (11)  (17), 6, 1.069792173, 4.930207827
               20578, (2) (10289), 7, 1.069789441, 5.930210559
             20582, (2) (41) (251), 5, 1.069788075, 3.930211925
            20590, (2) (5) (29) (71), 4, 1.069785345, 2.930214655
             20594, (2) (7) (1471), 3, 1.069783980, 1.930216020
               20602, (2) (10301), 5, 1.069781252, 3.930218748
                         2                                    
               20612, (2)  (5153), 4, 1.000000000, 3.000000000
                            2                                   
             20618, (2) (13)  (61), 3, 0.8115746631, 2.188425337
               20626, (2) (10313), 6, 1.069773073, 4.930226927
             20630, (2) (5) (2063), 4, 1.069771712, 2.930228288
             20638, (2) (17) (607), 6, 1.069768989, 4.930231011
               20642, (2) (10321), 5, 1.069767628, 3.930232372
                         2                                       
           20650, (2) (5)  (7) (59), 4, 0.9077758097, 3.092224190
             20654, (2) (23) (449), 8, 1.069763547, 6.930236453
               20662, (2) (10331), 3, 1.069760828, 1.930239172
               20666, (2) (10333), 3, 1.069759469, 1.930240531
               20674, (2) (10337), 8, 1.069756752, 6.930243248
               20686, (2) (10343), 3, 1.069752678, 1.930247322
             20690, (2) (5) (2069), 5, 1.069751321, 3.930248679
             20698, (2) (79) (131), 3, 1.069748608, 1.930251392
             20702, (2) (11) (941), 4, 1.311033099, 2.688966901
             20705, (5) (41) (101), 3, 1.000000000, 2.000000000
           20710, (2) (5) (19) (109), 4, 1.069744540, 2.930255460
               20714, (2) (10357), 3, 1.069743185, 1.930256815
             20722, (2) (13) (797), 4, 1.069740475, 2.930259525
             20726, (2) (43) (241), 5, 1.069739121, 3.930260879
             20734, (2) (7) (1481), 5, 1.069736413, 3.930263587
               20738, (2) (10369), 9, 1.069735060, 7.930264940
                     2                                           
           20740, (2)  (5) (17) (61), 3, 1.000000000, 2.000000000
           20746, (2) (11) (23) (41), 5, 1.310966664, 3.689033336
                           3                                   
             20750, (2) (5)  (83), 4, 0.7459102585, 3.254089742
             20758, (2) (97) (107), 6, 1.069728298, 4.930271702
            20770, (2) (5) (31) (67), 4, 1.069724244, 2.930275756
           20774, (2) (13) (17) (47), 6, 1.069722893, 4.930277107
               20782, (2) (10391), 3, 1.069720193, 1.930279807
             20794, (2) (37) (281), 5, 1.069716145, 3.930283855
             20806, (2) (101) (103), 4, 1.069712100, 2.930287900
             20810, (2) (5) (2081), 7, 1.069710752, 5.930289248
             20818, (2) (7) (1487), 3, 1.069708058, 1.930291942
             20822, (2) (29) (359), 4, 1.069706711, 2.930293289
                       2                                        
             20828, (2)  (41) (127), 3, 1.000000000, 2.000000000
             20830, (2) (5) (2083), 4, 1.069704018, 2.930295982
             20842, (2) (17) (613), 6, 1.069699982, 4.930300018
             20846, (2) (7) (1489), 5, 1.069698637, 3.930301363
                      2                                         
            20852, (2)  (13) (401), 3, 0.9999999999, 2.000000000
               20854, (2) (10427), 3, 1.069695948, 1.930304052
               20866, (2) (10433), 9, 1.069691916, 7.930308084
             20870, (2) (5) (2087), 4, 1.069690573, 2.930309427
                       2                                        
             20876, (2)  (17) (307), 3, 1.000000000, 2.000000000
           20878, (2) (11) (13) (73), 4, 1.310768371, 2.689231629
             20882, (2) (53) (197), 5, 1.069686546, 3.930313454
             20890, (2) (5) (2089), 5, 1.069683862, 3.930316138
             20894, (2) (31) (337), 5, 1.069682521, 3.930317479
             20902, (2) (7) (1493), 4, 1.069679840, 2.930320160
               20914, (2) (10457), 6, 1.069675820, 4.930324180
               20926, (2) (10463), 3, 1.069671802, 1.930328198
          20930, (2) (5) (7) (13) (23), 4, 1.069670464, 2.930329536
                            2                                   
             20938, (2) (19)  (29), 4, 0.7737240757, 3.226275924
                           2                                    
             20950, (2) (5)  (419), 4, 0.9079094975, 3.092090502
               20954, (2) (10477), 3, 1.069662439, 1.930337561
             20962, (2) (47) (223), 3, 1.069659767, 1.930340233
             20966, (2) (11) (953), 5, 1.310637010, 3.689362990
               20974, (2) (10487), 3, 1.069655761, 1.930344239
             20978, (2) (17) (617), 6, 1.069654426, 4.930345574
                      2                                         
            20980, (2)  (5) (1049), 3, 0.9999999999, 2.000000000
             20986, (2) (7) (1499), 3, 1.069651757, 1.930348243
             20990, (2) (5) (2099), 4, 1.069650423, 2.930349577
               20998, (2) (10499), 3, 1.069647756, 1.930352244
                       2                                       
             21004, (2)  (59) (89), 3, 1.000000000, 2.000000000
           21010, (2) (5) (11) (191), 4, 1.310571578, 2.689428422
             21022, (2) (23) (457), 4, 1.069639763, 2.930360237
               21026, (2) (10513), 5, 1.069638432, 3.930361568
             21034, (2) (13) (809), 5, 1.069635770, 3.930364230
              21035, (5) (7) (601), 3, 1.000000000, 2.000000000
             21046, (2) (17) (619), 6, 1.069631781, 4.930368219
                           2                                    
             21050, (2) (5)  (421), 4, 0.9079535499, 3.092046450
               21058, (2) (10529), 8, 1.069627794, 6.930372206
                             2                                   
           21070, (2) (5) (7)  (43), 4, 0.8741650653, 3.125834935
             21074, (2) (41) (257), 10, 1.069622482, 8.930377518
             21082, (2) (83) (127), 3, 1.069619828, 1.930380172
             21086, (2) (13) (811), 3, 1.069618501, 1.930381499
                         2                                    
               21092, (2)  (5273), 3, 1.000000000, 2.000000000
             21094, (2) (53) (199), 4, 1.069615849, 2.930384151
           21098, (2) (7) (11) (137), 5, 1.310441206, 3.689558794
             21106, (2) (61) (173), 4, 1.069611873, 2.930388127
             21110, (2) (5) (2111), 4, 1.069610548, 2.930389452
               21118, (2) (10559), 3, 1.069607899, 1.930392101
             21122, (2) (59) (179), 3, 1.069606576, 1.930393424
                         2                                    
               21124, (2)  (5281), 3, 1.000000000, 2.000000000
             21130, (2) (5) (2113), 7, 1.069603928, 5.930396072
                     2                                           
           21140, (2)  (5) (7) (151), 3, 1.000000000, 2.000000000
                                 2                              
             21142, (2) (11) (31) , 3, 0.9655643971, 2.034435603
             21146, (2) (97) (109), 6, 1.069598638, 4.930401362
                       2                                        
             21148, (2)  (17) (311), 3, 1.000000000, 2.000000000
             21154, (2) (7) (1511), 3, 1.069595995, 1.930404005
             21158, (2) (71) (149), 4, 1.069594674, 2.930405326
             21166, (2) (19) (557), 4, 1.069592032, 2.930407968
            21170, (2) (5) (29) (73), 5, 1.069590712, 3.930409288
               21178, (2) (10589), 5, 1.069588073, 3.930411927
            21182, (2) (7) (17) (89), 6, 1.069586753, 4.930413247
                         2                                    
               21188, (2)  (5297), 3, 1.000000000, 2.000000000
           21190, (2) (5) (13) (163), 4, 1.069584115, 2.930415885
               21202, (2) (10601), 6, 1.069580161, 4.930419839
             21206, (2) (23) (461), 4, 1.069578843, 2.930421157
               21214, (2) (10607), 3, 1.069576209, 1.930423791
               21226, (2) (10613), 5, 1.069572260, 3.930427740
           21230, (2) (5) (11) (193), 6, 1.310246867, 4.689753133
            21238, (2) (7) (37) (41), 5, 1.069568313, 3.930431687
           21242, (2) (13) (19) (43), 3, 1.069566998, 1.930433002
                       2                                        
             21244, (2)  (47) (113), 3, 1.000000000, 2.000000000
                           4                                   
             21250, (2) (5)  (17), 6, 0.5849939779, 5.415006022
                        3                                     
              21256, (2)  (2657), 3, 0.9304376015, 2.069562398
               21262, (2) (10631), 3, 1.069560428, 1.930439572
                       2                                        
             21284, (2)  (17) (313), 3, 1.000000000, 2.000000000
             21286, (2) (29) (367), 4, 1.069552554, 2.930447446
             21290, (2) (5) (2129), 6, 1.069551243, 4.930448757
             21298, (2) (23) (463), 3, 1.069548621, 1.930451379
                       2                                       
             21308, (2)  (7) (761), 3, 1.000000000, 2.000000000
             21310, (2) (5) (2131), 4, 1.069544690, 2.930455310
               21314, (2) (10657), 7, 1.069543381, 5.930456619
             21322, (2) (7) (1523), 3, 1.069540762, 1.930459238
               21334, (2) (10667), 3, 1.069536837, 1.930463163
             21335, (5) (17) (251), 4, 0.9999999999, 3.000000000
             21338, (2) (47) (227), 3, 1.069535529, 1.930464471
             21346, (2) (13) (821), 4, 1.069532915, 2.930467085
                         2                                       
           21350, (2) (5)  (7) (61), 4, 0.9080842139, 3.091915786
                       2                                        
             21356, (2)  (19) (281), 3, 1.000000000, 2.000000000
             21358, (2) (59) (181), 3, 1.069528995, 1.930471005
             21370, (2) (5) (2137), 4, 1.069525078, 2.930474922
               21382, (2) (10691), 3, 1.069521163, 1.930478837
                            2                                   
             21386, (2) (17)  (37), 6, 0.7853600194, 5.214639981
             21394, (2) (19) (563), 3, 1.069517251, 1.930482749
             21398, (2) (13) (823), 3, 1.069515948, 1.930484052
             21410, (2) (5) (2141), 5, 1.069512039, 3.930487961
               21418, (2) (10709), 5, 1.069509435, 3.930490565
             21430, (2) (5) (2143), 4, 1.069505531, 2.930494469
                      2                                         
            21436, (2)  (23) (233), 3, 0.9999999999, 2.000000000
             21442, (2) (71) (151), 3, 1.069501630, 1.930498370
             21454, (2) (17) (631), 6, 1.069497731, 4.930502269
               21458, (2) (10729), 5, 1.069496432, 3.930503568
               21466, (2) (10733), 5, 1.069493834, 3.930506166
           21470, (2) (5) (19) (113), 6, 1.069492536, 4.930507464
               21478, (2) (10739), 3, 1.069489941, 1.930510059
             21482, (2) (23) (467), 3, 1.069488644, 1.930511356
                       2                                        
             21484, (2)  (41) (131), 3, 1.000000000, 2.000000000
            21490, (2) (5) (7) (307), 4, 1.069486050, 2.930513950
             21494, (2) (11) (977), 6, 1.309862507, 4.690137493
             21502, (2) (13) (827), 3, 1.069482162, 1.930517838
              21506, (2) (10753), 10, 1.069480866, 8.930519134
                        3                                     
              21512, (2)  (2689), 3, 0.9305210768, 2.069478923
             21514, (2) (31) (347), 3, 1.069478276, 1.930521724
            21518, (2) (7) (29) (53), 4, 1.069476981, 2.930523019
             21526, (2) (47) (229), 3, 1.069474393, 1.930525607
             21530, (2) (5) (2153), 5, 1.069473099, 3.930526901
                       2                                       
             21532, (2)  (7) (769), 3, 1.000000000, 2.000000000
                            2                                  
             21538, (2) (11)  (89), 5, 1.069470512, 3.930529488
                           2                                    
             21550, (2) (5)  (431), 4, 0.9081701047, 3.091829895
             21554, (2) (13) (829), 3, 1.069465342, 1.930534658
                       2                                        
             21556, (2)  (17) (317), 3, 1.000000000, 2.000000000
               21562, (2) (10781), 5, 1.069462759, 3.930537241
             21566, (2) (41) (263), 5, 1.069461467, 3.930538533
                         2                                    
               21572, (2)  (5393), 3, 1.000000000, 2.000000000
            21574, (2) (7) (23) (67), 3, 1.069458886, 1.930541114
             21586, (2) (43) (251), 3, 1.069455016, 1.930544984
           21590, (2) (5) (17) (127), 6, 1.069453726, 4.930546274
               21598, (2) (10799), 3, 1.069451148, 1.930548852
             21610, (2) (5) (2161), 5, 1.069447283, 3.930552717
             21614, (2) (101) (107), 4, 1.069445995, 2.930554005
             21622, (2) (19) (569), 5, 1.069443420, 3.930556580
             21634, (2) (29) (373), 4, 1.069439560, 2.930560440
             21646, (2) (79) (137), 5, 1.069435703, 3.930564297
                           2                                    
             21650, (2) (5)  (433), 4, 0.9082126922, 3.091787308
                         2                                        
           21658, (2) (7)  (13) (17), 6, 0.8745120072, 5.125487993
                         2                                    
               21668, (2)  (5417), 3, 1.000000000, 2.000000000
           21670, (2) (5) (11) (197), 4, 1.309609401, 2.690390599
             21682, (2) (37) (293), 4, 1.069424146, 2.930575854
               21694, (2) (10847), 3, 1.069420299, 1.930579701
               21706, (2) (10853), 5, 1.069416455, 3.930583545
           21710, (2) (5) (13) (167), 4, 1.069415174, 2.930584826
             21715, (5) (43) (101), 3, 1.000000000, 2.000000000
                       2                                       
             21716, (2)  (61) (89), 3, 1.000000000, 2.000000000
               21718, (2) (10859), 3, 1.069412613, 1.930587387
               21722, (2) (10861), 3, 1.069411333, 1.930588667
            21730, (2) (5) (41) (53), 5, 1.069408774, 3.930591226
             21742, (2) (7) (1553), 6, 1.069404936, 4.930595064
             21746, (2) (83) (131), 3, 1.069403658, 1.930596342
             21754, (2) (73) (149), 5, 1.069401102, 3.930598898
                        2                                     
              21764, (2)  (5441), 4, 0.9999999999, 3.000000000
               21766, (2) (10883), 3, 1.069397270, 1.930602730
            21770, (2) (5) (7) (311), 4, 1.069395994, 2.930604006
               21778, (2) (10889), 6, 1.069393441, 4.930606559
             21790, (2) (5) (2179), 4, 1.069389614, 2.930610386
             21794, (2) (17) (641), 9, 1.069388339, 7.930611661
                     2                                           
           21812, (2)  (7) (19) (41), 3, 1.000000000, 2.000000000
             21814, (2) (13) (839), 3, 1.069381968, 1.930618032
             21826, (2) (7) (1559), 3, 1.069378149, 1.930621851
            21830, (2) (5) (37) (59), 4, 1.069376876, 2.930623124
             21838, (2) (61) (179), 3, 1.069374332, 1.930625668
                         2                                        
           21850, (2) (5)  (19) (23), 4, 0.9082971628, 3.091702837
             21862, (2) (17) (643), 6, 1.069366707, 4.930633293
                                 2                              
             21866, (2) (13) (29) , 4, 0.7323894635, 3.267610536
               21874, (2) (10937), 6, 1.069362898, 4.930637102
             21886, (2) (31) (353), 7, 1.069359091, 5.930640909
           21890, (2) (5) (11) (199), 4, 1.309296467, 2.690703533
             21895, (5) (29) (151), 3, 1.000000000, 2.000000000
               21898, (2) (10949), 5, 1.069355287, 3.930644713
             21902, (2) (47) (233), 5, 1.069354019, 3.930645981
            21910, (2) (5) (7) (313), 4, 1.069351485, 2.930648515
               21914, (2) (10957), 3, 1.069350218, 1.930649782
             21922, (2) (97) (113), 7, 1.069347686, 5.930652314
             21926, (2) (19) (577), 6, 1.069346420, 4.930653580
                      2                                         
            21940, (2)  (5) (1097), 3, 0.9999999999, 2.000000000
               21946, (2) (10973), 5, 1.069340095, 3.930659905
                           2                                    
             21950, (2) (5)  (439), 4, 0.9083390509, 3.091660949
               21958, (2) (10979), 3, 1.069336304, 1.930663696
               21959, (7) (3137), 3, 1.000000000, 2.000000000
                      2     2                                   
            21964, (2)  (17)  (19), 3, 0.7165981803, 2.283401820
                                3                              
             21970, (2) (5) (13) , 4, 0.5562109331, 3.443789067
             21982, (2) (29) (379), 4, 1.069328728, 2.930671272
               21986, (2) (10993), 6, 1.069327466, 4.930672534
             21994, (2) (7) (1571), 3, 1.069324944, 1.930675056
             21998, (2) (17) (647), 6, 1.069323683, 4.930676317
               22006, (2) (11003), 3, 1.069321162, 1.930678838
            22010, (2) (5) (31) (71), 4, 1.069319902, 2.930680098
             22018, (2) (101) (109), 4, 1.069317383, 2.930682617
                              2                                  
           22022, (2) (7) (11)  (13), 3, 1.069316123, 1.930683877
                        3                                     
              22024, (2)  (2753), 3, 0.9306845059, 2.069315494
             22030, (2) (5) (2203), 4, 1.069313606, 2.930686394
             22034, (2) (23) (479), 3, 1.069312348, 1.930687652
             22042, (2) (103) (107), 3, 1.069309832, 1.930690168
             22046, (2) (73) (151), 4, 1.069308574, 2.930691426
               22054, (2) (11027), 3, 1.069306060, 1.930693940
             22058, (2) (41) (269), 5, 1.069304803, 3.930695197
                       4                                        
             22064, (2)  (7) (197), 3, 0.8613941630, 2.138605837
           22066, (2) (11) (17) (59), 6, 1.309048826, 4.690951174
             22070, (2) (5) (2207), 4, 1.069301034, 2.930698966
            22078, (2) (7) (19) (83), 3, 1.069298523, 1.930701477
             22082, (2) (61) (181), 3, 1.069297268, 1.930702732
                                2                              
             22090, (2) (5) (47) , 4, 0.6843908624, 3.315609138
                    2    2                                        
          22100, (2)  (5)  (13) (17), 3, 0.8391098313, 2.160890169
             22102, (2) (43) (257), 10, 1.069290997, 8.930709003
               22114, (2) (11057), 7, 1.069287238, 5.930712762
           22126, (2) (13) (23) (37), 3, 1.069283480, 1.930716520
             22130, (2) (5) (2213), 5, 1.069282229, 3.930717771
               22138, (2) (11069), 5, 1.069279726, 3.930720274
                      2    2                                    
            22148, (2)  (7)  (113), 3, 0.8055159994, 2.194484001
                           2                                    
             22150, (2) (5)  (443), 4, 0.9084221438, 3.091577856
           22154, (2) (11) (19) (53), 4, 1.308925891, 2.691074109
             22162, (2) (7) (1583), 3, 1.069272224, 1.930727776
                       2                                        
             22172, (2)  (23) (241), 3, 1.000000000, 2.000000000
               22174, (2) (11087), 3, 1.069268476, 1.930731524
             22178, (2) (13) (853), 3, 1.069267227, 1.930732773
               22186, (2) (11093), 5, 1.069264731, 3.930735269
            22190, (2) (5) (7) (317), 4, 1.069263483, 2.930736517
             22198, (2) (11) (1009), 5, 1.308864645, 3.691135355
             22202, (2) (17) (653), 6, 1.069259742, 4.930740258
             22210, (2) (5) (2221), 4, 1.069257249, 2.930742751
             22214, (2) (29) (383), 4, 1.069256003, 2.930743997
             22222, (2) (41) (271), 5, 1.069253511, 3.930746489
               22226, (2) (11113), 3, 1.069252266, 1.930747734
               22234, (2) (11117), 5, 1.069249776, 3.930750224
                           2                                    
             22246, (2) (7)  (227), 3, 0.8748478224, 2.125152178
                           3                                   
             22250, (2) (5)  (89), 5, 0.7476819082, 4.252318092
               22255, (5) (4451), 3, 1.000000000, 2.000000000
             22258, (2) (31) (359), 3, 1.069242313, 1.930757687
           22270, (2) (5) (17) (131), 6, 1.069238585, 4.930761415
                         2                                    
               22276, (2)  (5569), 4, 1.000000000, 3.000000000
             22282, (2) (13) (857), 5, 1.069234859, 3.930765141
             22286, (2) (11) (1013), 4, 1.308742585, 2.691257415
             22294, (2) (71) (157), 3, 1.069231137, 1.930768863
               22298, (2) (11149), 3, 1.069229895, 1.930770105
             22306, (2) (19) (587), 3, 1.069227416, 1.930772584
            22310, (2) (5) (23) (97), 6, 1.069226176, 4.930773824
               22318, (2) (11159), 3, 1.069223697, 1.930776303
               22322, (2) (11161), 5, 1.069222458, 3.930777542
          22330, (2) (5) (7) (11) (29), 4, 1.308681774, 2.691318226
             22334, (2) (13) (859), 3, 1.069218743, 1.930781257
               22342, (2) (11171), 3, 1.069216268, 1.930783732
               22354, (2) (11177), 6, 1.069212557, 4.930787443
             22358, (2) (7) (1597), 3, 1.069211320, 1.930788680
             22366, (2) (53) (211), 4, 1.069208848, 2.930791152
             22370, (2) (5) (2237), 5, 1.069207612, 3.930792388
                     2                                            
           22372, (2)  (7) (17) (47), 3, 0.9999999997, 2.000000000
             22378, (2) (67) (167), 3, 1.069205142, 1.930794858
                       2                                        
             22388, (2)  (29) (193), 3, 1.000000000, 2.000000000
             22390, (2) (5) (2239), 4, 1.069201437, 2.930798563
             22402, (2) (23) (487), 3, 1.069197736, 1.930802264
             22406, (2) (17) (659), 6, 1.069196502, 4.930803498
             22414, (2) (7) (1601), 8, 1.069194037, 6.930805963
               22426, (2) (11213), 5, 1.069190339, 3.930809661
             22430, (2) (5) (2243), 4, 1.069189108, 2.930810892
             22438, (2) (13) (863), 3, 1.069186645, 1.930813355
                           2                                    
             22442, (2) (7)  (229), 3, 0.8749574015, 2.125042598
                           2                                    
             22450, (2) (5)  (449), 8, 0.9085451110, 7.091454889
             22454, (2) (103) (109), 3, 1.069181723, 1.930818277
             22466, (2) (47) (239), 3, 1.069178034, 1.930821966
                       2                                        
             22468, (2)  (41) (137), 3, 1.000000000, 2.000000000
             22474, (2) (17) (661), 6, 1.069175576, 4.930824424
               22486, (2) (11243), 3, 1.069171891, 1.930828109
           22490, (2) (5) (13) (173), 4, 1.069170663, 2.930829337
             22498, (2) (7) (1607), 3, 1.069168208, 1.930831792
                       2                                        
             22508, (2)  (17) (331), 3, 1.000000000, 2.000000000
             22510, (2) (5) (2251), 4, 1.069164528, 2.930835472
               22514, (2) (11257), 5, 1.069163301, 3.930836699
               22522, (2) (11261), 5, 1.069160850, 3.930839150
             22526, (2) (7) (1609), 4, 1.069159624, 2.930840376
             22534, (2) (19) (593), 6, 1.069157174, 4.930842826
             22538, (2) (59) (191), 3, 1.069155950, 1.930844050
                        4                                     
              22544, (2)  (1409), 3, 0.8616917745, 2.138308226
               22546, (2) (11273), 6, 1.069153500, 4.930846500
                         2                                       
           22550, (2) (5)  (11) (41), 5, 1.147813236, 3.852186764
               22558, (2) (11279), 3, 1.069149830, 1.930850170
             22562, (2) (29) (389), 5, 1.069148607, 3.930851393
            22570, (2) (5) (37) (61), 4, 1.069146161, 2.930853839
             22582, (2) (7) (1613), 4, 1.069142495, 2.930857505
             22586, (2) (23) (491), 3, 1.069141274, 1.930858726
             22606, (2) (89) (127), 5, 1.069135169, 3.930864831
          22610, (2) (5) (7) (17) (19), 6, 1.069133949, 4.930866051
                       3                                        
             22616, (2)  (11) (257), 4, 1.170025722, 2.829974278
             22618, (2) (43) (263), 3, 1.069131510, 1.930868490
                        2                                     
              22628, (2)  (5657), 3, 0.9999999996, 2.000000000
            22630, (2) (5) (31) (73), 4, 1.069127853, 2.930872147
               22634, (2) (11317), 3, 1.069126635, 1.930873365
               22642, (2) (11321), 6, 1.069124199, 4.930875801
                            2                                   
             22646, (2) (13)  (67), 3, 0.8133375565, 2.186662444
                       2                                        
             22652, (2)  (7) (809), 3, 0.9999999997, 2.000000000
             22654, (2) (47) (241), 5, 1.069120546, 3.930879454
               22658, (2) (11329), 7, 1.069119330, 5.930880670
             22666, (2) (7) (1619), 3, 1.069116897, 1.930883103
             22670, (2) (5) (2267), 4, 1.069115681, 2.930884319
           22678, (2) (17) (23) (29), 6, 1.069113249, 4.930886751
             22690, (2) (5) (2269), 4, 1.069109603, 2.930890397
             22694, (2) (7) (1621), 3, 1.069108389, 1.930891611
               22702, (2) (11351), 3, 1.069105960, 1.930894040
               22706, (2) (11353), 5, 1.069104746, 3.930895254
             22714, (2) (41) (277), 5, 1.069102320, 3.930897680
             22726, (2) (11) (1033), 4, 1.308140846, 2.691859154
             22730, (2) (5) (2273), 7, 1.069097469, 5.930902531
               22738, (2) (11369), 6, 1.069095046, 4.930904954
             22742, (2) (83) (137), 5, 1.069093834, 3.930906166
                         3                                       
           22750, (2) (5)  (7) (13), 4, 0.7482408320, 3.251759168
             22762, (2) (19) (599), 3, 1.069087780, 1.930912220
             22774, (2) (59) (193), 6, 1.069084151, 4.930915849
                     2                                            
           22780, (2)  (5) (17) (67), 3, 0.9999999996, 2.000000000
              22786, (2) (11393), 10, 1.069080524, 8.930919476
            22790, (2) (5) (43) (53), 4, 1.069079316, 2.930920684
                       2                                        
             22796, (2)  (41) (139), 3, 1.000000000, 2.000000000
               22798, (2) (11399), 3, 1.069076900, 1.930923100
             22802, (2) (13) (877), 3, 1.069075692, 1.930924308
             22810, (2) (5) (2281), 5, 1.069073277, 3.930926723
           22814, (2) (11) (17) (61), 6, 1.308022174, 4.691977826
               22822, (2) (11411), 3, 1.069069657, 1.930930343
             22826, (2) (101) (113), 6, 1.069068450, 4.930931550
                           2                                    
             22834, (2) (7)  (233), 5, 0.8751731540, 4.124826846
             22838, (2) (19) (601), 3, 1.069064834, 1.930935166
               22846, (2) (11423), 3, 1.069062424, 1.930937576
                           2                                    
             22850, (2) (5)  (457), 5, 0.9087060337, 4.091293966
            22862, (2) (7) (23) (71), 3, 1.069057607, 1.930942393
             22870, (2) (5) (2287), 4, 1.069055199, 2.930944801
               22874, (2) (11437), 3, 1.069053997, 1.930946003
             22882, (2) (17) (673), 6, 1.069051592, 4.930948408
                       2                                       
             22892, (2)  (59) (97), 3, 1.000000000, 2.000000000
               22894, (2) (11447), 3, 1.069047984, 1.930952016
                               2                              
               22898, (2) (107) , 3, 0.6035701745, 2.396429826
             22906, (2) (13) (881), 6, 1.069044381, 4.930955619
            22910, (2) (5) (29) (79), 4, 1.069043179, 2.930956821
                      2                                         
            22916, (2)  (17) (337), 3, 0.9999999995, 2.000000000
             22918, (2) (7) (1637), 4, 1.069040778, 2.930959222
             22922, (2) (73) (157), 4, 1.069039578, 2.930960422
             22930, (2) (5) (2293), 4, 1.069037179, 2.930962821
               22942, (2) (11471), 3, 1.069033582, 1.930966418
           22946, (2) (7) (11) (149), 4, 1.307845192, 2.692154808
             22954, (2) (23) (499), 3, 1.069029987, 1.930970013
             22958, (2) (13) (883), 3, 1.069028789, 1.930971211
               22966, (2) (11483), 3, 1.069026394, 1.930973606
             22970, (2) (5) (2297), 5, 1.069025196, 3.930974804
               22978, (2) (11489), 8, 1.069022803, 6.930977197
                              2                                  
           22990, (2) (5) (11)  (19), 4, 1.069019215, 2.930980785
               22994, (2) (11497), 3, 1.069018020, 1.930981980
            23002, (2) (7) (31) (53), 4, 1.069015629, 2.930984371
             23014, (2) (37) (311), 3, 1.069012044, 1.930987956
             23018, (2) (17) (677), 6, 1.069010851, 4.930989149
                       2                                        
             23020, (2)  (5) (1151), 3, 1.000000000, 2.000000000
             23026, (2) (29) (397), 4, 1.069008464, 2.930991536
                             2                                   
           23030, (2) (5) (7)  (47), 4, 0.8752793711, 3.124720629
               23038, (2) (11519), 3, 1.069004884, 1.930995116
             23042, (2) (41) (281), 6, 1.069003691, 4.930996309
                           2                                    
             23050, (2) (5)  (461), 5, 0.9087852337, 4.091214766
                       2                                        
             23060, (2)  (5) (1153), 4, 1.000000000, 3.000000000
             23062, (2) (13) (887), 3, 1.068997732, 1.931002268
             23074, (2) (83) (139), 3, 1.068994160, 1.931005840
             23078, (2) (11) (1049), 5, 1.307669427, 3.692330573
            23086, (2) (7) (17) (97), 6, 1.068990589, 4.931009411
             23090, (2) (5) (2309), 5, 1.068989399, 3.931010601
               23098, (2) (11549), 5, 1.068987021, 3.931012979
             23110, (2) (5) (2311), 4, 1.068983455, 2.931016545
           23114, (2) (7) (13) (127), 3, 1.068982267, 1.931017733
             23134, (2) (43) (269), 4, 1.068976329, 2.931023671
             23138, (2) (23) (503), 3, 1.068975142, 1.931024858
                     2                                           
           23140, (2)  (5) (13) (89), 3, 1.000000000, 2.000000000
             23146, (2) (71) (163), 3, 1.068972770, 1.931027230
                           2                                    
             23150, (2) (5)  (463), 4, 0.9088245252, 3.091175475
               23158, (2) (11579), 3, 1.068969213, 1.931030787
             23162, (2) (37) (313), 3, 1.068968028, 1.931031972
            23170, (2) (5) (7) (331), 4, 1.068965658, 2.931034342
             23182, (2) (67) (173), 4, 1.068962106, 2.931037894
               23194, (2) (11597), 5, 1.068958555, 3.931041445
             23198, (2) (7) (1657), 3, 1.068957371, 1.931042629
                        2                                     
              23204, (2)  (5801), 3, 0.9999999997, 2.000000000
             23206, (2) (41) (283), 5, 1.068955007, 3.931044993
           23210, (2) (5) (11) (211), 4, 1.307494863, 2.692505137
           23218, (2) (13) (19) (47), 3, 1.068951460, 1.931048540
             23222, (2) (17) (683), 6, 1.068950279, 4.931049721
           23230, (2) (5) (23) (101), 4, 1.068947916, 2.931052084
               23234, (2) (11617), 3, 1.068946736, 1.931053264
               23242, (2) (11621), 5, 1.068944374, 3.931055626
             23246, (2) (59) (197), 4, 1.068943195, 2.931056805
               23255, (5) (4651), 3, 1.000000000, 2.000000000
             23258, (2) (29) (401), 6, 1.068939655, 4.931060345
               23266, (2) (11633), 7, 1.068937297, 5.931062703
           23270, (2) (5) (13) (179), 4, 1.068936119, 2.931063881
                      2          2                              
            23276, (2)  (11) (23) , 3, 0.9266448648, 2.073355135
             23278, (2) (103) (113), 6, 1.068933762, 4.931066238
           23290, (2) (5) (17) (137), 6, 1.068930230, 4.931069770
                      2    2                                    
            23300, (2)  (5)  (233), 3, 0.8399557954, 2.160044205
             23302, (2) (61) (191), 3, 1.068926699, 1.931073301
               23314, (2) (11657), 6, 1.068923170, 4.931076830
             23318, (2) (89) (131), 5, 1.068921995, 3.931078005
                       2    3                                   
             23324, (2)  (7)  (17), 3, 0.6130328985, 2.386967102
             23326, (2) (107) (109), 3, 1.068919644, 1.931080356
             23330, (2) (5) (2333), 5, 1.068918468, 3.931081532
             23338, (2) (7) (1667), 3, 1.068916120, 1.931083880
             23342, (2) (11) (1061), 4, 1.307321486, 2.692678514
                      2                                         
            23348, (2)  (13) (449), 4, 0.9999999996, 3.000000000
                           2                                    
             23350, (2) (5)  (467), 4, 0.9089025016, 3.091097498
               23354, (2) (11677), 3, 1.068911424, 1.931088576
               23362, (2) (11681), 8, 1.068909078, 6.931090922
             23366, (2) (7) (1669), 3, 1.068907905, 1.931092095
           23374, (2) (13) (29) (31), 4, 1.068905560, 2.931094440
             23390, (2) (5) (2339), 4, 1.068900873, 2.931099127
                        2                                     
              23396, (2)  (5849), 3, 0.9999999997, 2.000000000
               23398, (2) (11699), 3, 1.068898530, 1.931101470
               23402, (2) (11701), 3, 1.068897360, 1.931102640
             23405, (5) (31) (151), 3, 1.000000000, 2.000000000
             23410, (2) (5) (2341), 4, 1.068895020, 2.931104980
             23414, (2) (23) (509), 4, 1.068893850, 2.931106150
                           2                                    
             23422, (2) (7)  (239), 3, 0.8754885896, 2.124511410
           23426, (2) (13) (17) (53), 6, 1.068890341, 4.931109659
                        2                                     
              23428, (2)  (5857), 3, 0.9999999997, 2.000000000
               23434, (2) (11717), 5, 1.068888003, 3.931111997
             23443, (7) (17) (197), 3, 1.000000000, 2.000000000
             23446, (2) (19) (617), 5, 1.068884499, 3.931115501
                         2                                       
           23450, (2) (5)  (7) (67), 4, 0.9089411899, 3.091058810
             23458, (2) (37) (317), 4, 1.068880996, 2.931119004
             23470, (2) (5) (2347), 4, 1.068877495, 2.931122505
                            2                                  
             23474, (2) (11)  (97), 6, 1.068876330, 4.931123670
             23482, (2) (59) (199), 3, 1.068873997, 1.931126003
             23494, (2) (17) (691), 6, 1.068870501, 4.931129499
            23506, (2) (7) (23) (73), 4, 1.068867006, 2.931132994
             23510, (2) (5) (2351), 4, 1.068865843, 2.931134157
           23530, (2) (5) (13) (181), 4, 1.068860025, 2.931139975
                                2                              
             23534, (2) (7) (41) , 5, 0.6999439268, 4.300056073
             23542, (2) (79) (149), 4, 1.068856537, 2.931143463
             23546, (2) (61) (193), 6, 1.068855376, 4.931144624
              23554, (2) (11777), 12, 1.068853052, 10.93114695
                       2                                        
             23564, (2)  (43) (137), 3, 1.000000000, 2.000000000
               23566, (2) (11783), 3, 1.068849569, 1.931150431
             23570, (2) (5) (2357), 5, 1.068848407, 3.931151593
               23578, (2) (11789), 5, 1.068846087, 3.931153913
             23582, (2) (13) (907), 3, 1.068844927, 1.931155073
                       5                                       
             23584, (2)  (11) (67), 3, 1.031629270, 1.968370730
                        2                                     
              23588, (2)  (5897), 3, 0.9999999998, 2.000000000
            23590, (2) (5) (7) (337), 5, 1.068842608, 3.931157392
             23594, (2) (47) (251), 3, 1.068841449, 1.931158551
                      2                                         
            23596, (2)  (17) (347), 3, 0.9999999995, 2.000000000
               23602, (2) (11801), 6, 1.068839131, 4.931160869
           23606, (2) (11) (29) (37), 4, 1.306978230, 2.693021770
               23614, (2) (11807), 3, 1.068835656, 1.931164344
                           2                                    
             23618, (2) (7)  (241), 5, 0.8755916309, 4.124408369
               23626, (2) (11813), 5, 1.068832183, 3.931167817
           23630, (2) (5) (17) (139), 6, 1.068831026, 4.931168974
             23638, (2) (53) (223), 4, 1.068828712, 2.931171288
                       2                                        
             23644, (2)  (23) (257), 5, 1.000000000, 4.000000000
                         2                                       
           23650, (2) (5)  (11) (43), 4, 1.147114202, 2.852885798
               23662, (2) (11831), 3, 1.068821778, 1.931178222
                       2                                       
             23668, (2)  (61) (97), 3, 1.000000000, 2.000000000
            23674, (2) (7) (19) (89), 5, 1.068818313, 3.931181687
             23686, (2) (13) (911), 3, 1.068814850, 1.931185150
           23690, (2) (5) (23) (103), 4, 1.068813697, 2.931186303
                            2                                   
             23698, (2) (17)  (41), 6, 0.7875473878, 5.212452612
             23710, (2) (5) (2371), 4, 1.068807932, 2.931192068
             23714, (2) (71) (167), 3, 1.068806780, 1.931193220
             23722, (2) (29) (409), 5, 1.068804476, 3.931195524
                      2                                         
            23732, (2)  (17) (349), 3, 0.9999999998, 2.000000000
               23734, (2) (11867), 3, 1.068801023, 1.931198977
             23735, (5) (47) (101), 3, 1.000000000, 2.000000000
             23746, (2) (31) (383), 3, 1.068797571, 1.931202429
                           4                                   
             23750, (2) (5)  (19), 4, 0.5895753933, 3.410424607
               23755, (5) (4751), 4, 1.000000000, 3.000000000
             23758, (2) (7) (1697), 7, 1.068794122, 5.931205878
                               2                              
               23762, (2) (109) , 3, 0.6031894575, 2.396810542
             23770, (2) (5) (2377), 5, 1.068790674, 3.931209326
                     2                                           
           23780, (2)  (5) (29) (41), 3, 1.000000000, 2.000000000
               23794, (2) (11897), 6, 1.068783784, 4.931216216
             23798, (2) (73) (163), 4, 1.068782638, 2.931217362
               23806, (2) (11903), 3, 1.068780343, 1.931219657
             23810, (2) (5) (2381), 5, 1.068779197, 3.931220803
                         2                                    
               23812, (2)  (5953), 4, 1.000000000, 3.000000000
               23818, (2) (11909), 5, 1.068776904, 3.931223096
             23822, (2) (43) (277), 3, 1.068775758, 1.931224242
             23830, (2) (5) (2383), 4, 1.068773467, 2.931226533
             23834, (2) (17) (701), 6, 1.068772322, 4.931227678
           23842, (2) (7) (13) (131), 3, 1.068770032, 1.931229968
             23845, (5) (19) (251), 4, 1.000000000, 3.000000000
                       2                                       
             23852, (2)  (67) (89), 3, 1.000000000, 2.000000000
               23854, (2) (11927), 3, 1.068766598, 1.931233402
                      2                                         
            23860, (2)  (5) (1193), 3, 0.9999999995, 2.000000000
               23866, (2) (11933), 5, 1.068763168, 3.931236832
          23870, (2) (5) (7) (11) (31), 4, 1.306639546, 2.693360454
               23878, (2) (11939), 3, 1.068759739, 1.931240261
             23890, (2) (5) (2389), 4, 1.068756312, 2.931243688
             23894, (2) (13) (919), 3, 1.068755170, 1.931244830
           23902, (2) (17) (19) (37), 6, 1.068752887, 4.931247113
               23906, (2) (11953), 5, 1.068751746, 3.931248254
             23926, (2) (7) (1709), 4, 1.068746044, 2.931253956
             23930, (2) (5) (2393), 5, 1.068744904, 3.931255096
                       2                                        
             23932, (2)  (31) (193), 3, 1.000000000, 2.000000000
               23938, (2) (11969), 9, 1.068742626, 7.931257374
                           2                                    
             23950, (2) (5)  (479), 4, 0.9091317091, 3.090868291
            23954, (2) (7) (29) (59), 4, 1.068738071, 2.931261929
                       2                                        
             23956, (2)  (53) (113), 3, 1.000000000, 2.000000000
               23962, (2) (11981), 5, 1.068735794, 3.931264206
             23966, (2) (23) (521), 5, 1.068734656, 3.931265344
               23969, (11) (2179), 4, 1.237779892, 2.762220108
               23974, (2) (11987), 3, 1.068732382, 1.931267618
             23986, (2) (67) (179), 3, 1.068728971, 1.931271029
             23990, (2) (5) (2399), 4, 1.068727834, 2.931272166
                       2                                       
             23996, (2)  (7) (857), 3, 1.000000000, 2.000000000
                            2                                   
             23998, (2) (13)  (71), 3, 0.8144107606, 2.185589239
                                     627

>