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

%I A304412 #24 May 06 2025 12:48:28
%S A304412 1,6,8,9,12,48,16,12,12,72,24,72,28,96,96,15,36,72,40,108,128,144,48,
%T A304412 96,18,168,16,144,60,576,64,18,192,216,192,108,76,240,224,144,84,768,
%U A304412 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
%N A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).
%H A304412 Andrew Howroyd, <a href="/A304412/b304412.txt">Table of n, a(n) for n = 1..1000</a>
%H A304412 Ilya Gutkovskiy, <a href="/A304412/a304412.jpg">Scatter plot of a(n) up to n=50000</a>.
%H A304412 Vaclav Kotesovec, <a href="/A304412/a304412_1.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%H A304412 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A304412 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A304412 a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
%F A304412 a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
%F A304412 a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
%F A304412 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - _Amiram Eldar_, Sep 17 2023
%F A304412 From _Vaclav Kotesovec_, May 06 2025: (Start)
%F A304412 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).
%F A304412 Dirichlet g.f.: zeta(s-1)^2 * f(s).
%F A304412 Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
%F A304412 f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
%F A304412 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...
%F A304412 and gamma is the Euler-Mascheroni constant A001620. (End)
%e A304412 a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.
%t A304412 a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
%t A304412 Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]
%o A304412 (PARI) a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ _Andrew Howroyd_, Jul 24 2018
%Y A304412 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.
%Y A304412 Cf. A065463.
%K A304412 nonn,easy,mult
%O A304412 1,2
%A A304412 _Ilya Gutkovskiy_, May 12 2018