A343498 -id:A343498 - cp's OEIS frontend

A343514 a(n) = Sum_{k=1..n} (k/gcd(n, k))^4.

1, 2, 18, 84, 355, 645, 2276, 3192, 7413, 9400, 25334, 18395, 60711, 52747, 88760, 106688, 243849, 137790, 432346, 275570, 499867, 522513, 1151404, 561415, 1542125, 1214436, 1907502, 1569673, 3756719, 1344999, 5274000, 3451216, 4970577, 4690778, 7499154, 4217504, 12948595, 8207261, 11565572

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

Cf. A053820, A057661, A332654, A343498, A343513.

Table[Sum[(k/GCD[n, k])^4, {k, 1, n}], {n, 1, 39}]

a(n) = sum(k=1, n, (k/gcd(n, k))^4); \\ Michel Marcus, Apr 17 2021

a(n) = Sum_{d|n} A053820(d).

A343518 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= n} gcd(x_1, x_2, x_3 , x_4, n).

Original entry on oeis.org

1, 6, 17, 42, 74, 153, 216, 379, 531, 809, 1011, 1605, 1832, 2626, 3268, 4304, 4861, 6798, 7333, 9878, 11148, 13711, 14972, 19985, 20775, 25643, 28503, 34517, 35988, 46162, 46406, 57092, 61077, 70986, 75099, 92520, 91426, 108693, 115774, 135491, 135791, 165719, 163227, 193437

Offset: 1

Seiichi Manyama, Apr 17 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Column 4 of A343516.

Cf. A000010, A343498.

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 3, 4] &]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)

a(n) = sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, gcd(gcd(gcd(gcd(n, a), b), c), d)))));

a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+3, 4));

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

a(n) = Sum_{d|n} phi(n/d) * binomial(d+3, 4).
G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^5.
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (10800*zeta(5)). - Vaclav Kotesovec, May 23 2021

A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

Original entry on oeis.org

1, 47, 323, 1744, 3749, 15181, 19207, 59648, 84969, 176203, 175691, 563312, 399853, 902729, 1210927, 1970176, 1503377, 3993543, 2606419, 6538256, 6203861, 8257477, 6716183, 19266304, 12105625, 18793091, 21172347, 33497008, 21218429, 56913569, 29552671, 64028672

Offset: 1

Seiichi Manyama, May 17 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A343498, A372926, A372929, A372931.
Cf. A000203, A008683.
Cf. A013661, A013664, A157292.

f[p_, e_] := p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

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

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^4.
a(n) = Sum_{d|n} mu(n/d) * d^4 * sigma(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, c = zeta(2)/zeta(6) = 315/(2*Pi^4) = 1.616892... (A157292). (End)
Mobius transformation of A280022. - R. J. Mathar, Jul 14 2025

cp's OEIS Frontend

A343514 a(n) = Sum_{k=1..n} (k/gcd(n, k))^4.

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A343518 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= n} gcd(x_1, x_2, x_3 , x_4, n).

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A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

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