A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99

Offset: 1

Author

Ilya Gutkovskiy, Sep 13 2019

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A002131.

Dirichlet convolution of A000027 with A001227.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

Programs

Mathematica

nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]

PARI

a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).

G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.

a(n) = Sum_{d|n} A002131(d).

a(n) = Sum_{d|n} d * A001227(n/d).

a(n) = (A007429(n) + A288417(n)) / 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022