A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.

1, 4, 4, 8, 4, 16, 4, 12, 8, 16, 4, 32, 4, 16, 16, 16, 4, 32, 4, 32, 16, 16, 4, 48, 8, 16, 12, 32, 4, 64, 4, 20, 16, 16, 16, 64, 4, 16, 16, 48, 4, 64, 4, 32, 32, 16, 4, 64, 8, 32, 16, 32, 4, 48, 16, 48, 16, 16, 4, 128, 4, 16, 32, 24, 16, 64, 4, 32, 16, 64, 4

Offset: 1

Author

Werner Schulte, Dec 03 2018

Keywords

• nonn
• easy
• mult

Comments

Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see A322327, for k = 3 see A226602 (offset 1), and for k = 4 see this sequence.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000007, A001221, A005361, A007427, A008836, A034444, A226602, A322327.

Programs

Maple

f:= n -> mul(4*t[2],t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Dec 07 2018

Mathematica

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

PARI

a(n) = my(f=factor(n)); vecprod(f[,2])*4^omega(n); \\ Michel Marcus, Dec 04 2018

Python

from math import prod
from sympy import factorint
def A322328(n): return prod(e<<2 for e in factorint(n).values()) # Chai Wah Wu, Dec 24 2022

Formula

Multiplicative with a(p^e) = 4*e for prime p and e > 0.

Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.

Dirichlet inverse is b(n) = a(n) * A008836(n) for n > 0, and b(n) is multiplicative with b(p^e) = 4*e*(-1)^e for prime p and e > 0.

Equals Dirichlet convolution of A034444 with itself.

Equals Dirichlet convolution of A000005 with abs(A007427).