A317831 Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

1, 3, 2, 19, 3, 3, 4, 63, 9, 9, 6, 19, 7, 6, 6, 867, 9, 27, 10, 57, 8, 9, 12, 63, 11, 21, 11, 19, 15, 9, 16, 3069, 12, 27, 12, 171, 19, 15, 14, 189, 21, 12, 22, 57, 27, 18, 24, 867, 41, 33, 18, 133, 27, 33, 18, 63, 20, 45, 30, 57, 31, 24, 18, 22199, 21, 18, 34, 171, 24, 18, 36, 567, 37, 57, 22, 95, 24, 21, 40, 2601, 227, 63, 42, 19, 27, 33

Offset: 1

Author

Antti Karttunen, Aug 10 2018

Keywords

• nonn
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A317832 (gives the denominators).

Cf. also A000203, A299151.

Programs

PARI

A317831perA317832(n) = if(1==n,n,(sigma(n)-sumdiv(n,d,if((d>1)&&(dA317831perA317832(d)*A317831perA317832(n/d),0)))/2);
A317831(n) = numerator(A317831perA317832(n));

PARI

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

Formula

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

Sum_{k=1..n} A317831(k) / A317832(k) ~ n^2 * sqrt(Pi/(24*log(n))) * (1 - (gamma - 1 + 6*zeta'(2)/Pi^2) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025