A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Author

Ilya Gutkovskiy, May 12 2018

Keywords

• nonn
• easy
• mult

Examples

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

Programs

Mathematica

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

PARI

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).

a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.

a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023

From Vaclav Kotesovec, May 06 2025: (Start)

Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).

Dirichlet g.f.: zeta(s-1)^2 * f(s).

Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where

f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,

f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...

and gamma is the Euler-Mascheroni constant A001620. (End)