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

> ```