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%I #18 Sep 17 2023 10:00:29
%S 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,
%T 104,12,84,58,240,62,12,132,136,140,54,74,152,156,80,82,336,86,132,90,
%U 184,94,60,21,60,204,156,106,48,220,112,228,232,118,360,122,248,126,14,260
%N If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).
%H Andrew Howroyd, <a href="/A304409/b304409.txt">Table of n, a(n) for n = 1..1000</a>
%H Ilya Gutkovskiy, <a href="/A304409/a304409.jpg">Scatter plot of a(n) up to n=50000</a>.
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F a(n) = A000005(n)*A007947(n).
%F a(p^k) = p*(k + 1) where p is a prime and k > 0.
%F a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
%F 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
%e a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.
%t a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
%t Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]
%o (PARI) a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ _Andrew Howroyd_, Jul 24 2018
%Y 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.
%K nonn,easy,mult
%O 1,2
%A _Ilya Gutkovskiy_, May 12 2018
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