A260591 a(n) is the number of odd numbers k < 2^n such that A260590(k) = n.

0, 1, 0, 1, 2, 0, 3, 7, 0, 12, 0, 30, 85, 0, 173, 476, 0, 961, 0, 2652, 8045, 0, 17637, 51033, 0, 108950, 312455, 0, 663535, 0, 1900470, 5936673, 0, 13472296, 39993895, 0, 87986917, 0, 257978502, 820236724, 0, 1899474678, 5723030586, 0, 12809477536, 38036848410, 0, 84141805077, 0, 248369601964

Offset: 1

Author

Joseph K. Horn, O. Praem, and Robert G. Wilson v, Jul 29 2015

Keywords

• nonn

Comments

a(n) is either 0 or about c^(n-1) with c = log(3)/log(2).

Nonzero values give A100982. - Ruud H.G. van Tol, Nov 25 2021

A close variant of this sequence, that starts at offset 0, but with a(0)=0 and a(1)=1, maps it to the count of dropping patterns of 2^n+c(2^n), with the c(2^n) as mentioned with A177789. The positions of the zeros of that variant sequence might be a close variant of A054414, again with a(0)=0 (not properly checked yet). - Ruud H.G. van Tol, Nov 28 2021

It appears that the proportion of zeros is 1-log(2)/log(3) = 36.907...%. - Jesse Randall, Oct 10 2024

Examples

			a(1) = 0 since there exists no odd number whose msa is 1;
a(2) = 1 since there is only one odd number, 5 with k=2 2k+1, with k less than 2^2 whose msa is 2;
a(3) = 0 since there exists no odd number whose msa is 3;
a(4) = 1 since there is only one number, 1, less than 2^(4+1) whose msa is 4;
a(5) = 2 since there are two numbers, 11 & 23, less than 2^(4+1) whose msa is 4; etc.
		

Crossrefs

Cf. A260590, A054414, A100982, A186009, A186008, A177789, A346640.

Programs

Mathematica

msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; g[n_] := Length@ Select[ Range[ 2^(n - 1)], f@# == n &]; Array[ g, 20]

Extensions

a(31) onwards from Jesse Randall, Sep 09 2024