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

%I A304409 #22 Jun 06 2025 12:35:38
%S A304409 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 A304409 104,12,84,58,240,62,12,132,136,140,54,74,152,156,80,82,336,86,132,90,
%U A304409 184,94,60,21,60,204,156,106,48,220,112,228,232,118,360,122,248,126,14,260
%N A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).
%H A304409 Andrew Howroyd, <a href="/A304409/b304409.txt">Table of n, a(n) for n = 1..1000</a>
%H A304409 Ilya Gutkovskiy, <a href="/A304409/a304409.jpg">Scatter plot of a(n) up to n=50000</a>.
%H A304409 Vaclav Kotesovec, <a href="/A304409/a304409_1.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%H A304409 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A304409 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A304409 a(n) = A000005(n)*A007947(n).
%F A304409 a(p^k) = p*(k + 1) where p is a prime and k > 0.
%F A304409 a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
%F A304409 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
%F A304409 From _Vaclav Kotesovec_, Jun 06 2025: (Start)
%F A304409 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.
%F A304409 Dirichlet g.f.: zeta(s-1)^2 * f(s).
%F A304409 Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
%F A304409 f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
%F A304409 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...
%F A304409 and gamma is the Euler-Mascheroni constant A001620. (End)
%e A304409 a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.
%t A304409 a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
%t A304409 Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]
%o A304409 (PARI) a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ _Andrew Howroyd_, Jul 24 2018
%Y A304409 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 A304409 nonn,easy,mult
%O A304409 1,2
%A A304409 _Ilya Gutkovskiy_, May 12 2018