A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 1024, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1

Offset

1,2

Comments

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

a(n) = 2^A318455(n).

Sum_{k=1..n} A318453(k) / a(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

Mathematica

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

Prog

(PARI)
up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318454(n) = denominator(v318453_54[n]);

Crossrefs

Cf. A001227.

Cf. A318453 (numerators), A318455.

Sequence in context: A317832 A317928 A011327 * A317926 A318318 A318314

Adjacent sequences: A318451 A318452 A318453 * A318455 A318456 A318457

Keyword

nonn,frac

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved