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A322328
a(n) = A005361(n) * 4^A001221(n) for n > 0.
10
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
1,2
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
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).
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 *)
PROG
(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
KEYWORD
nonn,easy,mult
AUTHOR
Werner Schulte, Dec 03 2018
STATUS
approved