A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260

Offset: 1

Author

Ilya Gutkovskiy, May 12 2018

Keywords

• nonn
• easy
• mult

Examples

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.
		

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 (100000 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, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

Programs

Mathematica

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]

PARI

a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

Formula

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

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

a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.

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

From Vaclav Kotesovec, Jun 06 2025: (Start)

Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(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) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,

f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...

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