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

-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, 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, 2, 1, 2, 2, 3, 3, 4, 3, 5, 1, 5, 3, 2, 1, 3, 2, 3, 4, 3

Offset: 1

Author

Nicholas M. R. Frieler, Aug 19 2024

Keywords

• base
• sign

Comments

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.

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.

Examples

			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.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Kevin Ryde, C Code

Crossrefs

Cf. A121805, A374725.

Programs

C

/* See links. */

Mathematica

ExponentialCommaSequenceLength[n_] := Module[{seq = {n}, i = 1},
  While[True,
  Do[
    If[(IntegerDigits@Power[Last@seq, Mod[Last@commaSeq,10]*10 + j])[[1]] == j,
      seq = seq~Join~{Power[Last@seq, Mod[Last@commaSeq, 10]*10 + j]};
      Break[];];,
     {j, 1, 9}
    ];
   If[Length@seq != i + 1, Break[];];
   If[seq[[1]] == seq[[2]], Return[-1]];
   i++;
 ];
  Length@seq
 ]