A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741, 4776, 4929, 7101, 6877, 9546, 10121, 12069, 12189, 17284, 16145, 19881, 21321, 25826, 24417, 33669, 29821, 38224, 38889, 44361, 45021, 58386, 50689, 61893, 64061, 76884, 68961, 91089, 79549, 99234, 101781

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} phi(n/d) * d^3.

a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).

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

Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021

Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - Vaclav Kotesovec, May 20 2021

Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - Amiram Eldar, Nov 22 2022

a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 20 2024

Maple

with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024

Mathematica

a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, gcd(k, n)^3);
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4))

Crossrefs

Column 3 of A343510.

Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

Sequence in context: A273113 A273177 A055195 * A273243 A273297 A273268

Adjacent sequences: A343494 A343495 A343496 * A343498 A343499 A343500

Keyword

nonn,mult,easy

Author

Seiichi Manyama, Apr 17 2021

Status

approved