A317832 Denominators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

1, 2, 1, 8, 1, 1, 1, 16, 2, 2, 1, 4, 1, 1, 1, 128, 1, 4, 1, 8, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 256, 1, 2, 1, 16, 1, 1, 1, 16, 1, 1, 1, 4, 2, 1, 1, 64, 2, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 1, 1024, 1, 1, 1, 8, 1, 1, 1, 32, 1, 2, 1, 4, 1, 1, 1, 128, 8, 2, 1, 1, 1, 1, 1, 8, 1, 4, 1, 2, 1, 1, 1, 128, 1, 4, 1, 8, 1, 1, 1, 16, 1

Offset: 1

Author

Antti Karttunen, Aug 10 2018

Keywords

• nonn
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A317831 (gives the numerators).

Cf. also A000203, A299152.

Programs

PARI

A317831perA317832(n) = if(1==n,n,(sigma(n)-sumdiv(n,d,if((d>1)&&(dA317831perA317832(d)*A317831perA317832(n/d),0)))/2);
A317832(n) = denominator(A317831perA317832(n));

PARI

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/((1-p*X)*(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000203(n) - Sum_{d|n, d>1, d 1.