A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.

0, 1, 1, 1024, 1, 60073, 1, 1048576, 59049, 9766649, 1, 61514752, 1, 282476273, 9824674, 1073741824, 1, 3547250577, 1, 10001048576, 282534298, 25937425625, 1, 62991106048, 9765625, 137858492873, 3486784401, 289255703552, 1, 586710856801, 1, 1099511627776, 25937483650

Offset: 1

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

• nonn

Comments

Dirichlet convolution of A010051(n) and n^10. - Wesley Ivan Hurt, Jul 15 2025

Examples

			a(6) = 60073; a(6) = 6^10 * Sum_{p|6, p prime} 1/p^10 = 60466176 * (1/2^10 + 1/3^10) = 60073.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

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), A351249 (k=9), this sequence (k=10).

Cf. A000040, A001221, A010051, A069095.

Programs

Maple

f:= proc(n) local p;
  n^10 * add(1/p^10, p = numtheory:-factorset(n))
end proc:
map(f, [$1..40]); # Robert Israel, Sep 10 2024

Mathematica

Join[{0},Table[n^10 Total[1/FactorInteger[n][[;;,1]]^10],{n,2,40}]] (* Harvey P. Dale, Aug 10 2024 *)

PARI

a(n) = my(f=factor(n)); n^10*sum(k=1, #f~, 1/f[k,1]^10); \\ Michel Marcus, Sep 10 2024

Python

from sympy import primefactors
def A351262(n): return sum((n//p)**10 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

Formula

a(A000040(n)) = 1.

a(n) = Sum_{d|n} A069095(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)^10, where c = A010051.

a(p^k) = p^(10*k-10) for p prime and k>=1. (End)