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A304412
If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).
6
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
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
KEYWORD
nonn,easy,mult
AUTHOR
Ilya Gutkovskiy, May 12 2018
STATUS
approved