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

1,2

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)

Formula

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

Prog

(PARI)
A317831perA317832(n) = if(1==n, n, (sigma(n)-sumdiv(n, d, if((d>1)&&(d<n), A317831perA317832(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

Crossrefs

Cf. A317832 (gives the denominators).

Cf. also A000203, A299151.

Sequence in context: A006281 A157705 A185447 * A078073 A317927 A075568

Adjacent sequences: A317828 A317829 A317830 * A317832 A317833 A317834

Keyword

nonn,frac

Author

Antti Karttunen, Aug 10 2018

Status

approved