A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809

Offset: 1

Author

Seiichi Manyama, Apr 17 2021

Keywords

• nonn
• mult
• easy

Links

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

Crossrefs

Column 5 of A343510.

Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

Programs

Magma

A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

Mathematica

a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); 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)^5);

PARI

a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))

SageMath

def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

Formula

a(n) = Sum_{d|n} phi(n/d) * d^5.

a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).

G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.

Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021

Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021

Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022

a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024