A322327 a(n) = A005361(n) * A034444(n).

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, 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, 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

Offset: 1

Author

Werner Schulte, Dec 03 2018

Keywords

• nonn
• easy
• mult

Comments

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.

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

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

Programs

Mathematica

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

PARI

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

PARI

A322327(n) = factorback(apply(e -> e+e, factor(n)[, 2])); \\ Antti Karttunen, Jul 18 2022

Python

from math import prod
from sympy import factorint
def A322327(n): return prod(e<<1 for e in factorint(n).values()) # Chai Wah Wu, Dec 26 2022

Formula

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

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

Equals Dirichlet convolution of A000005 and A227291.

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

a(n) = A005361(n^2) = A286324(n^2). - Amiram Eldar, Dec 09 2023

Extensions

Data section extended up to a(100) by Antti Karttunen, Jul 18 2022