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

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.

Index entries for sequences computed from exponents in factorization of n.

Index entries for sequences computed from indices in prime factorization.

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

Example

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

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}]

Prog

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

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.

Sequence in context: A305704 A043720 A191980 * A161982 A285275 A117096

Adjacent sequences: A304409 A304410 A304411 * A304413 A304414 A304415

Keyword

nonn,easy,mult

Author

Ilya Gutkovskiy, May 12 2018

Status

approved