A351198 Sum of the 10th powers of the primes dividing n.

0, 1024, 59049, 1024, 9765625, 60073, 282475249, 1024, 59049, 9766649, 25937424601, 60073, 137858491849, 282476273, 9824674, 1024, 2015993900449, 60073, 6131066257801, 9766649, 282534298, 25937425625, 41426511213649, 60073, 9765625, 137858492873, 59049, 282476273

Offset

1,2

Comments

Inverse Möbius transform of n^10 * 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^10.

G.f.: Sum_{k>=1} prime(k)^10 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 16 2022

Additive with a(p^e) = p^10. - Amiram Eldar, Jun 20 2022

a(n) = Sum_{d|n} d^10 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Mathematica

Array[DivisorSum[#, #^10 &, PrimeQ] &, 50]
f[p_, e_] := p^10; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

Prog

(Python)
from sympy import primefactors
def A351198(n): return sum(p**10 for p in primefactors(n)) # Chai Wah Wu, Feb 04 2022
(PARI) a(n) = vecsum(apply(x->x^10, factor(n)[, 1])); \\ Michel Marcus, 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), A351196 (k=8), A351197 (k=9), this sequence (k=10).

Cf. A010051, A030629 (p^10).

Sequence in context: A182681 A182682 A305941 * A195250 A016901 A017684

Adjacent sequences: A351195 A351196 A351197 * A351199 A351200 A351201

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved