A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

1, 3, 4, 6, 4, 12, 4, 10, 10, 12, 4, 24, 4, 12, 16, 15, 4, 30, 4, 24, 16, 12, 4, 40, 10, 12, 20, 24, 4, 48, 4, 21, 16, 12, 16, 60, 4, 12, 16, 40, 4, 48, 4, 24, 40, 12, 4, 60, 10, 30, 16, 24, 4, 60, 16, 40, 16, 12, 4, 96, 4, 12, 40, 28, 16, 48, 4, 24, 16, 48, 4, 100, 4, 12, 40

Offset: 1

Author

Ilya Gutkovskiy, Oct 18 2019

Keywords

• nonn
• mult

Comments

Inverse Moebius transform applied twice to A001227.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001227, A007425, A007426, A133700, A280486, A318845.

Programs

Mathematica

Table[Sum[DivisorSigma[0, n/d] Total[Mod[Divisors[d], 2]], {d, Divisors[n]}], {n, 1, 75}]
nmax = 75; A007425 = Table[DivisorSum[n, DivisorSigma[0, #] &], {n, 1, nmax}]; Table[DivisorSum[n, A007425[[#]] &, OddQ[n/#] &], {n, 1, nmax}]
f[2, e_] := (e + 1)*(e + 2)/2; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

G.f.: Sum_{k>=1} tau_3(k) * x^k / (1 - x^(2*k)), where tau_3 = A007425.

a(n) = tau_4(n) if n odd, tau_4(n) - tau_4(n/2) if n even, where tau_4 = A007426.

a(n) = Sum_{d|n, n/d odd} tau_3(d).

a(n) = Sum_{d|n} A000005(n/d) * A001227(d).

Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A280486.

Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (e+1)*(e+2)*(e+3)/6 for odd primes p. - Amiram Eldar, Dec 02 2020