A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.

0, 1, 1, 256, 1, 6817, 1, 65536, 6561, 390881, 1, 1745152, 1, 5765057, 397186, 16777216, 1, 44726337, 1, 100065536, 5771362, 214359137, 1, 446758912, 390625, 815730977, 43046721, 1475854592, 1, 2664570241, 1, 4294967296, 214365442, 6975757697, 6155426, 11449942272

Offset: 1

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

• nonn

Comments

Dirichlet convolution of A010051(n) and n^8. - Wesley Ivan Hurt, Jul 15 2025

Examples

			a(6) = 6817; a(6) = 6^8 * Sum_{p|6, p prime} 1/p^8 = 1679616 * (1/2^8 + 1/3^8) = 6817.
		

Crossrefs

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), this sequence (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069093.

Programs

Mathematica

Array[#^8*DivisorSum[#, 1/#^8 &, PrimeQ] &, 36] (* Stefano Spezia, Jul 15 2025 *)

Python

from sympy import primefactors
def A351248(n): return sum((n//p)**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

Formula

a(A000040(n)) = 1.

a(n) = Sum_{d|n} A069093(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025

From Wesley Ivan Hurt, Jul 15 2025: (Start)

a(n) = Sum_{d|n} c(d) * (n/d)^8, where c = A010051.

a(p^k) = p^(8*k-8) for p prime and k>=1. (End)