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

1,2

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

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

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 *)

Prog

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

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.

Sequence in context: A283576 A203284 A321832 * A016900 A017680 A210840

Adjacent sequences: A351193 A351194 A351195 * A351197 A351198 A351199

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved