A343498 - OEIS

A343498

a(n) = Sum_{k=1..n} gcd(k, n)^4.

1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207, 70216, 83537, 114393, 130339, 172346, 199781, 249067, 279863, 364204, 393145, 485741, 545067, 659518, 707309, 887519, 923551, 1123472, 1216033, 1420129, 1514003, 1843746, 1874197

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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^4.

a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).

G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.

Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021

Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - Vaclav Kotesovec, May 20 2021

Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - Amiram Eldar, Nov 22 2022

a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 18 2024

MATHEMATICA

a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)

f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); 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)^4);

(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 3));

(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))

(Magma)

A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;

[A343498(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

(SageMath)

def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))

[A343498(n) for n in range(1, 51)] # G. C. Greubel, Jun 24 2024

CROSSREFS