A384816 Sum of the cubes of the indices of distinct prime factors of n.

0, 1, 8, 1, 27, 9, 64, 1, 8, 28, 125, 9, 216, 65, 35, 1, 343, 9, 512, 28, 72, 126, 729, 9, 27, 217, 8, 65, 1000, 36, 1331, 1, 133, 344, 91, 9, 1728, 513, 224, 28, 2197, 73, 2744, 126, 35, 730, 3375, 9, 64, 28, 351, 217, 4096, 9, 152, 65, 520, 1001, 4913, 36, 5832, 1332, 72, 1, 243, 134, 6859, 344, 737, 92

Offset: 1

Author

Ilya Gutkovskiy, Jun 10 2025

Keywords

• nonn

Links

• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000720, A005064, A066328, A332385.

Programs

Mathematica

Table[Plus @@ (PrimePi[#[[1]]]^3 & /@ FactorInteger[n]), {n, 70}]
nmax = 70; CoefficientList[Series[Sum[k^3 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = my(f=factor(n)[,1]); sum(k=1, #f~, primepi(f[k])^3); \\ Michel Marcus, Jun 10 2025

Formula

If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^3), where pi = A000720.

G.f.: Sum_{k>=1} k^3 * x^prime(k) / (1 - x^prime(k)).