A322327 - OEIS

%I #33 Jul 27 2024 15:56:15

%S 1,2,2,4,2,4,2,6,4,4,2,8,2,4,4,8,2,8,2,8,4,4,2,12,4,4,6,8,2,8,2,10,4,

%T 4,4,16,2,4,4,12,2,8,2,8,8,4,2,16,4,8,4,8,2,12,4,12,4,4,2,16,2,4,8,12,

%U 4,8,2,8,4,8,2,24,2,4,8,8,4,8,2,16,8,4,2,16,4,4,4,12,2,16,4,8,4,4,4,20,2,8,8,16

%N a(n) = A005361(n) * A034444(n).

%C Conjecture: 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 this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.

%C Dirichlet inverse b(n) [= A355837(n)] is multiplicative with b(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e > 0.

%H Antti Karttunen, <a href="/A322327/b322327.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F Multiplicative with a(p^e) = 2*e for prime p and e > 0.

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

%F Equals Dirichlet convolution of A000005 and A227291.

%F Sum_{k=1..n} a(k) ~ 15*(log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 - 360*zeta'(4)/Pi^4) * n / Pi^2 + 6*zeta(1/2)^2 * sqrt(n) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 20 2020

%F a(n) = A005361(n^2) = A286324(n^2). - _Amiram Eldar_, Dec 09 2023

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

%o (PARI) a(n) = my(f=factor(n)); vecprod(f[,2])*2^omega(n); \\ _Michel Marcus_, Dec 04 2018

%o (PARI) A322327(n) = factorback(apply(e -> e+e, factor(n)[, 2])); \\ _Antti Karttunen_, Jul 18 2022

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A322327(n): return prod(e<<1 for e in factorint(n).values()) # _Chai Wah Wu_, Dec 26 2022

%Y Cf. A000005, A000007, A001221, A001620, A005361, A034444, A226602, A227291, A286324, A322328, A355837 (Dirichlet inverse).

%K nonn,easy,mult

%O 1,2

%A _Werner Schulte_, Dec 03 2018

%E Data section extended up to a(100) by _Antti Karttunen_, Jul 18 2022