A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

1, 3, 4, 8, 6, 12, 8, 18, 14, 18, 12, 32, 14, 24, 24, 39, 18, 42, 20, 48, 32, 36, 24, 72, 32, 42, 44, 64, 30, 72, 32, 81, 48, 54, 48, 112, 38, 60, 56, 108, 42, 96, 44, 96, 84, 72, 48, 156, 58, 96, 72, 112, 54, 132, 72, 144, 80, 90, 60, 192, 62, 96, 112, 166, 84

Offset

1,2

Links

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

Formula

G.f.: Sum_{k>=1} sigma(k) * (theta_3(x^k) - 1) / 2.

a(n) = Sum_{d|n} A076752(d).

a(n) = Sum_{d|n} A206369(n/d) * tau(d).

a(n) = Sum_{d|n} A010052(n/d) * sigma(d).

a(n) = Sum_{d|n} A124315(n/d) * phi(d).

a(n) = Sum_{d|n} A046951(n/d) * d.

a(p) = p + 1, where p is prime.

Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1080. - Vaclav Kotesovec, Feb 26 2020

Mathematica

Table[Sum[Boole[IntegerQ[(n/d)^(1/2)]] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] (EllipticTheta[3, 0, x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) A332844(n) = sumdiv(n, d, issquare(n/d) * sigma(d)); \\ Antti Karttunen, May 23 2021

Crossrefs

Cf. A000005, A000010, A000203, A010052, A046951, A076752 (Mobius transf.), A124315, A206369, A344442, A347090 (Dirichlet inverse).

Sequence in context: A074212 A125715 A129283 * A347228 A330575 A079787

Adjacent sequences: A332841 A332842 A332843 * A332845 A332846 A332847

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 26 2020

Status

approved