LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Strings of Consecutive Happy Numbers 23 Feb 20081 Mar 200823 March 200925 Nov 200927 Nov 2009The goal here is to show that the first string of six consecutive happy numbers begins with 7899999999999959999999996 which has S(7899999999999959999999996) = 1739. Amazingly, this string continues to give seven in a row. Note that Dr. Grundman uses S_2 but we will simply use S in the explanation, which in our program is the procedure 'onestep'. Also note we will use a dot as the digit concatenation operator. Thus, 111112999994 could be written 11111.2.9999.94 if we wish. First we define our procedures. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=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The next procedure is only needed when we want to find the smallest N with S(N) = n for a given n. A separate worksheet has the details on how this is constructed. The array contains the smallest N for 1 <= n <= 486 = 6*81. 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first verify that 7899999999999959999999996 does indeed give seven happy numbers in a row. 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 now check to see if there is a smaller string of 6 consecutive happy numbers with the first one having the last two digits less than 94 (so no carry to the third digit.) Since our known solution has 25 digits with first digit 7, we need only check numbers N with S(N) < 7^2+25*9^2 = 1993. Because we need to worry about carries later, we need to consider up to 1993+(9^2 - 8^2) =2010. We split N into N1*10^2 + b where b is a two digit number. Then S(N) = S(N1) + S(b). Let a = S(N1). So we need only check if a + S(b) is happy. 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So we never have more than 3 in a row with first digit zero. 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so we conclude that there are no four in a row with last digit, 0, 1, 2, 3, 4, 5 and for last digit 6, there is only one value of M1 with four in a row. We next check for d0=7 for three in a row. 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Actually, we only needed to go up to 2025, not 2500, so we can ignore the 2493 entry. Note that 1839 works for both d=6 and d=7; of course, d=6 is better. So we may assume N ends with a 6 or 7, for now assume it is 7. Set P = N+2 = N1.9=N2.d.99...99.9 and let k be the number of digits of 9 at the end of N1 (the ellipsis part). Let M2 = S(N2). Certainly M2< M1-9^2<2000. Then P+1 = N2.(d+1).00...00.0 so S(P+1) = S(N2). Also, S(P+2) = S(N2)+1^2 and S(P+3) = S(N2)+2^2, etc. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElaXF1b0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkjbW5HRiQ2JFEkNTY4RicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2JFEjODFGJ0Y+LyUrZXhlY3V0YWJsZUdGRkY+Rj4tRkE2LVEiO0YnRj5GREZHRklGS0ZNRk9GUUZTL0ZXUSwwLjI3Nzc3NzhlbUYnLUZBNi1RIn5GJ0Y+RkQvRkhGRkZJRktGTUZPRlFGUy9GV0ZVRmZuRj4=First consider M1=568. Set N = N1.9 where S(N1) = 568. Set N1 = N2.d.9...9 where d is a digit not equal 9, and there are exactly k digits of nine ending N1. Clearly, k*81 cannot exceed 568 so k is at most 7. Then N+3 = N2.(d+1).0...0.00.We have S(N1) = S(N2)+d^2+k*81 = 568.We have S(N+3) = S(N2) +(d+1)^2 = S(N1) +2d+1 - k*81 = 568 + 2d+1-k*81We will check all possible k values from 0 to 7 and all possible digits from 0 to 8 to see if
S(N+3) = 568 + 2d+1-k*81S(N+4) = 568 + 2d+1-k*81 + 1^2S(N+5) = 568 + 2d+1-k*81 + 2^2are each happy. 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JSFHWe see that the smallest possibility is k=9, d=5, and S(N2) = 1085. Here L(1085) = 78999999999999. Thus, the number N is the concatenation N2.d.9...99.6 which is 78999999999999.5.999999999.6. JSFH