A321294 a(n) = Sum_{d|n} mu(n/d)*d*sigma_n(d).

1, 9, 83, 1058, 15629, 282381, 5764807, 134480900, 3486902505, 100048836321, 3138428376731, 107006403495850, 3937376385699301, 155572843119518781, 6568408661060858767, 295150157013526773768, 14063084452067724991025, 708236697425777157039381

Offset: 1

Author

Ilya Gutkovskiy, Nov 02 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..385

Crossrefs

Cf. A018804, A069097, A320940, A332517, A342432, A342433.

Programs

Mathematica

Table[Sum[MoebiusMu[n/d] d DivisorSigma[n, d], {d, Divisors[n]}], {n, 18}]
Table[Sum[EulerPhi[n/d] d^(n + 1), {d, Divisors[n]}], {n, 18}]
Table[Sum[GCD[n, k]^(n + 1), {k, n}], {n, 18}]

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n)); \\ Michel Marcus, Nov 03 2018

Python

from sympy import totient, divisors
def A321294(n):
    return sum(totient(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = [x^n] Sum_{i>=1} Sum_{j>=1} mu(i)*j^(n+1)*x^(i*j)/(1 - x^(i*j))^2.

a(n) = Sum_{d|n} phi(n/d)*d^(n+1).

a(n) = Sum_{k=1..n} gcd(n,k)^(n+1).

a(n) ~ n^(n+1). - Vaclav Kotesovec, Nov 02 2018