A384815 Sum of the cubes of the exponents in the prime factorization of n.

0, 1, 1, 8, 1, 2, 1, 27, 8, 2, 1, 9, 1, 2, 2, 64, 1, 9, 1, 9, 2, 2, 1, 28, 8, 2, 27, 9, 1, 3, 1, 125, 2, 2, 2, 16, 1, 2, 2, 28, 1, 3, 1, 9, 9, 2, 1, 65, 8, 9, 2, 9, 1, 28, 2, 28, 2, 2, 1, 10, 1, 2, 9, 216, 2, 3, 1, 9, 2, 3, 1, 35, 1, 2, 9, 9, 2, 3, 1, 65, 64, 2, 1, 10, 2, 2, 2, 28, 1, 10

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

R. L. Duncan, A class of additive arithmetical functions, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.

Index entries for sequences computed from exponents in factorization of n.

Formula

If n = Product (p_j^k_j) then a(n) = Sum (k_j^3).

From Amiram Eldar, Jul 03 2025: (Start)

Additive with a(p^e) = e^3.

Sum_{k=1..n} a(k) ~ n * log(log(n)) + B_3 * n + O(n/log(n)), where B_3 = gamma + Sum_{p prime} ((1-1/p)*Sum_{m>=1} m^3/p^m + log(1-1/p)) = 16.17021843694072992072..., and gamma is Euler's constant (A001620) (Duncan, 1962). (End)

Mathematica

Join[{0}, Table[Plus @@ (#[[2]]^3 & /@ FactorInteger[n]), {n, 2, 90}]]

Prog

(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 2]^3); \\ Michel Marcus, Jun 10 2025

Crossrefs

Cf. A001222, A001620, A005064, A090885, A360970.

Sequence in context: A225767 A019763 A307506 * A387150 A179050 A176458

Adjacent sequences: A384812 A384813 A384814 * A384816 A384817 A384818

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jun 10 2025

Status

approved