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A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

%I A304411 #17 Nov 30 2022 08:34:46
%S A304411 1,3,4,6,6,12,8,9,8,18,12,24,14,24,24,12,18,24,20,36,32,36,24,36,12,
%T A304411 42,12,48,30,72,32,15,48,54,48,48,38,60,56,54,42,96,44,72,48,72,48,48,
%U A304411 16,36,72,84,54,36,72,72,80,90,60,144,62,96,64,18,84,144,68,108,96,144,72,72
%N A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).
%H A304411 Andrew Howroyd, <a href="/A304411/b304411.txt">Table of n, a(n) for n = 1..1000</a>
%H A304411 Ilya Gutkovskiy, <a href="/A304411/a304411.jpg">Scatter plot of a(n) up to n=50000 </a>.
%H A304411 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A304411 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A304411 a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
%F A304411 a(p^k) = (p + 1)*k where p is a prime and k > 0.
%F A304411 a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
%F A304411 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - _Amiram Eldar_, Nov 30 2022
%e A304411 a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.
%t A304411 a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
%t A304411 Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]
%o A304411 (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ _Andrew Howroyd_, Jul 24 2018
%Y A304411 Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.
%Y A304411 Cf. A072691, A330596.
%K A304411 nonn,mult
%O A304411 1,2
%A A304411 _Ilya Gutkovskiy_, May 12 2018