A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.

1, 129, 2189, 16514, 78129, 282381, 823549, 2113796, 4787349, 10078641, 19487181, 36149146, 62748529, 106237821, 171024381, 270565896, 410338689, 617568021, 893871757, 1290222306, 1802748761, 2513846349, 3404825469, 4627099444, 6103828145, 8094560241

Offset: 1

Author

Seiichi Manyama, Apr 17 2021

Keywords

• nonn
• mult
• easy

Comments

In general, for m > 1, if a(n) = Sum_{j=1..n} gcd(j, n)^m, then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1) * zeta(m+1)). - Vaclav Kotesovec, May 20 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Column 7 of A343510.

Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.

Programs

Mathematica

a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI

a(n) = sum(k=1, n, gcd(k, n)^7);

PARI

a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+120*x^k+1191*x^(2*k)+2416*x^(3*k)+1191*x^(4*k)+120*x^(5*k)+x^(6*k))/(1-x^k)^8))

Formula

a(n) = Sum_{d|n} phi(n/d) * d^7.

a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).

G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8.

Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021

Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - Vaclav Kotesovec, May 20 2021

Multiplicative with a(p^e) = p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1). - Amiram Eldar, Nov 22 2022

a(n) = Sum_{1 <= i_1, ..., i_7 <= n} gcd(i_1, ..., i_7, n) = Sum_{d divides n} d * J_7(n/d), where the Jordan totient function J_7(n) = A069092(n). - Peter Bala, Jan 29 2024