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A375567 Length of the "exponential comma sequence" with n as the initial term, or -1 if that sequence is infinite.

%I A375567 #24 Feb 01 2025 08:44:53
%S A375567 -1,3,4,1,2,3,5,4,4,-1,5,1,1,4,1,1,1,1,1,6,1,11,4,9,5,1,7,2,3,1,1,1,6,
%T A375567 3,1,6,4,1,7,1,3,2,2,1,1,1,2,2,1,2,1,2,2,4,7,2,4,1,2,1,3,2,1,8,3,1,6,
%U A375567 2,1,2,2,3,3,4,3,5,1,5,3,2,1,3,2,3,4,3
%N A375567 Length of the "exponential comma sequence" with n as the initial term, or -1 if that sequence is infinite.
%C A375567 An "exponential comma sequence" is the lexicographically earliest sequence of positive integers (with some chosen initial term) with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive logarithms between the terms.
%C A375567 If the decimal expansion of n is 1...0, its exponential comma sequence is trivially constant and therefore infinite so that a(n) = -1. Conjecture: these are the only infinite exponential comma sequences.
%H A375567 Kevin Ryde, <a href="/A375567/b375567.txt">Table of n, a(n) for n = 1..10000</a>
%H A375567 Kevin Ryde, <a href="/A375567/a375567.c.txt">C Code</a>
%e A375567 For n = 2, the next term of its exponential comma sequence is 67108864 because log_2(67108864) = 26 and this is the smallest number where the exponential comma property holds.
%t A375567 ExponentialCommaSequenceLength[n_] := Module[{seq = {n}, i = 1},
%t A375567   While[True,
%t A375567   Do[
%t A375567     If[(IntegerDigits@Power[Last@seq, Mod[Last@commaSeq,10]*10 + j])[[1]] == j,
%t A375567       seq = seq~Join~{Power[Last@seq, Mod[Last@commaSeq, 10]*10 + j]};
%t A375567       Break[];];,
%t A375567      {j, 1, 9}
%t A375567     ];
%t A375567    If[Length@seq != i + 1, Break[];];
%t A375567    If[seq[[1]] == seq[[2]], Return[-1]];
%t A375567    i++;
%t A375567  ];
%t A375567   Length@seq
%t A375567  ]
%o A375567 (C) /* See links. */
%Y A375567 Cf. A121805, A374725.
%K A375567 base,sign
%O A375567 1,2
%A A375567 _Nicholas M. R. Frieler_, Aug 19 2024