A328854 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2 * p^(-s))^2.

1, 4, 4, 12, 4, 16, 4, 32, 12, 16, 4, 48, 4, 16, 16, 80, 4, 48, 4, 48, 16, 16, 4, 128, 12, 16, 32, 48, 4, 64, 4, 192, 16, 16, 16, 144, 4, 16, 16, 128, 4, 64, 4, 48, 48, 16, 4, 320, 12, 48, 16, 48, 4, 128, 16, 128, 16, 16, 4, 192, 4, 16, 48, 448, 16, 64, 4, 48, 16, 64, 4, 384, 4, 16, 48

Offset: 1

Author

Ilya Gutkovskiy, Oct 28 2019

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A061142 with itself.

Links

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

Crossrefs

Cf. A000005, A001222, A061142, A123667, A322328.

Programs

Mathematica

Table[2^PrimeOmega[n] DivisorSigma[0, n], {n, 1, 75}]
f[p_, e_] := (e + 1)*2^e; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

PARI

a(n) = numdiv(n)*2^bigomega(n); \\ Michel Marcus, Dec 02 2020

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

Formula

If n = Product (p_j^k_j) then a(n) = Product (2^k_j * (k_j + 1)).

a(n) = 2^bigomega(n) * tau(n), where bigomega = A001222 and tau = A000005.