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

Author

Ilya Gutkovskiy, Feb 26 2020

Keywords

• nonn
• mult

Links

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

Crossrefs

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

Programs

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
f[p_, e_] := (2*p^(e + 3) - e*p^2 + e - If[OddQ[e], 3*p^2 - 1, 2*p^2 + 2*p - 2])/(2*(p - 1)*(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, May 25 2025 *)

PARI

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

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

Multiplicative with a(p^e) = (2*p^(e+3) - e*p^2 + e - 3*p^2 + 1)/(2*(p-1)*(p^2-1)) if e is odd, and (2*p^(e+3) - e*p^2 + e - 2*p^2 - 2*p + 2)/(2*(p-1)*(p^2-1)) if e is even. - Amiram Eldar, May 25 2025