A351196 Sum of the 8th powers of the primes dividing n.

0, 256, 6561, 256, 390625, 6817, 5764801, 256, 6561, 390881, 214358881, 6817, 815730721, 5765057, 397186, 256, 6975757441, 6817, 16983563041, 390881, 5771362, 214359137, 78310985281, 6817, 390625, 815730977, 6561, 5765057, 500246412961, 397442, 852891037441, 256

Offset: 1

Author

Wesley Ivan Hurt, Feb 04 2022

Keywords

• nonn

Comments

Inverse Möbius transform of n^8 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Cf. A010051.

Programs

Mathematica

Array[DivisorSum[#, #^8 &, PrimeQ] &, 50]
f[p_, e_] := p^8; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

Python

from sympy import primefactors
def A351196(n): return sum(p**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

Formula

a(n) = Sum_{p|n, p prime} p^8.

G.f.: Sum_{k>=1} prime(k)^8 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022

Additive with a(p^e) = p^8. - Amiram Eldar, Jun 20 2022

a(n) = Sum_{d|n} d^8 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024