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A328975 Numbers whose trajectory under repeated application of the map in A053392 increases without limit.

%I A328975 #47 May 21 2025 19:04:20
%S A328975 1496,1497,1498,1499,1587,1588,1589,1691,1692,1693,1694,1695,1696,
%T A328975 1697,1698,1699,1719,1728,1729,1783,1784,1785,1786,1787,1788,1789,
%U A328975 1791,1792,1793,1794,1795,1796,1797,1798,1799,1819,1867,1868,1869,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885
%N A328975 Numbers whose trajectory under repeated application of the map in A053392 increases without limit.
%C A328975 Computed by _Hans Havermann_, Nov 01 2019.
%C A328975 There is a simple technique, due to _Hans Havermann_, that proves that many terms that in this sequence blow up: find a term in the trajectory that has an internal substring of three 9's. See A328974 for the case 1496.
%C A328975 The same reasoning shows that almost all numbers belong to the sequence.
%C A328975 From _Scott R. Shannon_, Nov 23 2019: (Start)
%C A328975 Although many of the numbers in this sequence eventually reach a term that contains three 9's, some do not. The first such example is 4949 which leads to 44444 after two steps, and starts a sequence of values that leads back to a much larger all-4's number every nine steps. No 9's appear in any of the intermediate values. Similar series found which lead to limitless loops are values with all 4's with a final 5, or all 3's with a final digit of 1 to 6. Of the first 33909 starting values that increase without limit 209 lead into one of these non-9 loops. (End)
%C A328975 From _Scott R. Shannon_, Nov 24 2019: (Start)
%C A328975 One can show that strings of repeated digits longer than a certain length will increase without limit by calculating the number of digits when a new value containing only the same digit appears again in the iterative sequence. Below shows the details of this for digits 1 to 9. The number of cycles is the number of iterations of A053392 required before a new value containing only the starting digit is seen. The minimum starting value is the smallest same-digit number such that subsequent iterations will produce another value with only the same digit of equal or longer length. The number of digits in the subsequence reoccurrence shows the number of digits in this value given the starting value has n digits. All sufficiently long single-digit numbers, with digits 1 to 8, increase 8-fold in length, minus a constant, after 9 iterations.
%C A328975 .
%C A328975 digit | # of cycles | min start value | # digits in reoccurrence
%C A328975   1   |      9      |     1111111     |      8*n - 45
%C A328975   2   |      9      |      222222     |      8*n - 38
%C A328975   3   |      3      |       33333     |      2*n -  5
%C A328975   4   |      9      |       44444     |      8*n - 31
%C A328975   5   |      9      |       55555     |      8*n - 33
%C A328975   6   |      3      |        6666     |      2*n -  4
%C A328975   7   |      9      |       77777     |      8*n - 27
%C A328975   8   |      9      |        8888     |      8*n - 28
%C A328975   9   |      2      |         999     |      2*n -  3
%C A328975 (End)
%H A328975 Scott R. Shannon, <a href="/A328975/b328975.txt">Table of n, a(n) for n = 1..33909</a> (terms 1..109 from Hans Havermann).
%H A328975 Hans Havermann, <a href="/A328975/a328975.txt">Proof of correctness of first 109 terms</a>
%Y A328975 Cf. A053392, A328974.
%K A328975 nonn,base
%O A328975 1,1
%A A328975 _N. J. A. Sloane_, Nov 02 2019