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

Author

N. J. A. Sloane

Keywords

• nonn

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

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).

Programs

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 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ Michel Marcus, Jul 11 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

Scheme

(define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

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

Extensions

More terms from Antti Karttunen, Jul 10 2017