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

%I A322328 #26 Dec 26 2022 15:48:25
%S A322328 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,
%T A322328 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,
%U A322328 4,48,16,48,16,16,4,128,4,16,32,24,16,64,4,32,16,64,4
%N A322328 a(n) = A005361(n) * 4^A001221(n) for n > 0.
%C A322328 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.
%H A322328 Robert Israel, <a href="/A322328/b322328.txt">Table of n, a(n) for n = 1..10000</a>
%F A322328 Multiplicative with a(p^e) = 4*e for prime p and e > 0.
%F A322328 Dirichlet g.f.: (zeta(s))^4 / (zeta(2*s))^2.
%F A322328 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.
%F A322328 Equals Dirichlet convolution of A034444 with itself.
%F A322328 Equals Dirichlet convolution of A000005 with abs(A007427).
%p A322328 f:= n -> mul(4*t[2],t=ifactors(n)[2]):
%p A322328 map(f, [$1..100]); # _Robert Israel_, Dec 07 2018
%t A322328 a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 4^Length[f] * Times@@f[[;; , 2]]]]; Array[a, 100] (* _Amiram Eldar_, Dec 03 2018 *)
%o A322328 (PARI) a(n) = my(f=factor(n)); vecprod(f[,2])*4^omega(n); \\ _Michel Marcus_, Dec 04 2018
%o A322328 (Python)
%o A322328 from math import prod
%o A322328 from sympy import factorint
%o A322328 def A322328(n): return prod(e<<2 for e in factorint(n).values()) # _Chai Wah Wu_, Dec 24 2022
%Y A322328 Cf. A000005, A000007, A001221, A005361, A007427, A008836, A034444, A226602, A322327.
%K A322328 nonn,easy,mult
%O A322328 1,2
%A A322328 _Werner Schulte_, Dec 03 2018