A005064 Sum of cubes of primes dividing n.

0, 8, 27, 8, 125, 35, 343, 8, 27, 133, 1331, 35, 2197, 351, 152, 8, 4913, 35, 6859, 133, 370, 1339, 12167, 35, 125, 2205, 27, 351, 24389, 160, 29791, 8, 1358, 4921, 468, 35, 50653, 6867, 2224, 133, 68921, 378, 79507, 1339, 152, 12175, 103823, 35, 343, 133, 4940, 2205, 148877, 35, 1456, 351, 6886, 24397, 205379, 160

Offset

1,2

Comments

The set of these terms is A213519. - Bernard Schott, Feb 11 2022

Inverse Möbius transform of n^3 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

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

G.f.: Sum_{k>=1} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016

From Antti Karttunen, Jul 11 2017: (Start)

a(n) = A005067(n) + 8*A059841(n).

a(n) = A005080(n) + A005084(n) + 8*A059841(n).

a(n) = A005072(n) + A005076(n) + 27*A079978(n).

(End)

Dirichlet g.f.: primezeta(s-3)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018

a(n) = Sum_{p|n, p prime} p^3. - Wesley Ivan Hurt, Feb 04 2022

a(n) = Sum_{d|n} d^3 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Mathematica

Array[DivisorSum[#, #^3 &, PrimeQ] &, 60] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := p^3; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

Prog

(Scheme) (define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017
(Python)
from sympy import primefactors
def a(n): return sum(p**3 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017
(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]^3); \\ Michel Marcus, Jul 11 2017

Crossrefs

Cf. A000578, A005067, A005072, A005076, A005080, A005084, A059841, A079978.

Cf. A010051, A213519.

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), this sequence (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Sequence in context: A070491 A070490 A088378 * A056551 A356193 A356192

Adjacent sequences: A005061 A005062 A005063 * A005065 A005066 A005067

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Antti Karttunen, Jul 10 2017

Status

approved