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

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, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72

Offset: 1

Author

Ilya Gutkovskiy, May 12 2018

Keywords

• nonn
• mult

Examples

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000 .
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

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.

Cf. A072691, A330596.

Programs

Mathematica

a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]

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

Formula

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).

a(p^k) = (p + 1)*k where p is a prime and k > 0.

a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).

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