A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

0, 1, 1, 512, 1, 20195, 1, 262144, 19683, 1953637, 1, 10339840, 1, 40354119, 1972808, 134217728, 1, 397498185, 1, 1000262144, 40373290, 2357948203, 1, 5293998080, 1953125, 10604499885, 387420489, 20661308928, 1, 39453437071, 1, 68719476736, 2357967374, 118587877009, 42306732

Offset: 1

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

• nonn

Comments

Dirichlet convolution of A010051(n) and n^9. - Wesley Ivan Hurt, Jul 15 2025

Examples

			a(6) = 20195; a(6) = 6^9 * Sum_{p|6, p prime} 1/p^9 = 10077696 * (1/2^9 + 1/3^9) = 20195.
		

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), A351248 (k=8), this sequence (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069094.

Programs

Mathematica

Array[#^9*DivisorSum[#, 1/#^9 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

Python

from sympy import primefactors
def A351249(n): return sum((n//p)**9 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

Formula

a(A000040(n)) = 1.

a(n) = Sum_{d|n} A069094(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)^9, where c = A010051.

a(p^k) = p^(9*k-9) for p prime and k>=1. (End)