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
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.
Programs
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Mathematica
Join[{0}, Table[Plus @@ (#[[2]]^3 & /@ FactorInteger[n]), {n, 2, 90}]] -
PARI
a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]^3); \\ Michel Marcus, Jun 10 2025
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)