A343497 -id:A343497 - cp's OEIS frontend

A343513 a(n) = Sum_{k=1..n} (k/gcd(n, k))^3.

1, 2, 10, 30, 101, 137, 442, 526, 1063, 1202, 3026, 1965, 6085, 4853, 7310, 8654, 18497, 10100, 29242, 17630, 29557, 30857, 64010, 30397, 77601, 60842, 89272, 71913, 164837, 60737, 216226, 139470, 188165, 180338, 265142, 152544, 443557, 282665, 371134, 275726, 672401, 251066, 815410, 461645

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

a(n) = 1+n^2*(n-1)^2/4 if n is prime. - Robert Israel, Apr 19 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053819, A057661, A332654, A343497, A343514.

f:= proc(n) local k;
  add((k/igcd(n,k))^3,k=1..n)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 19 2021

Table[Sum[(k/GCD[n, k])^3, {k, 1, n}], {n, 1, 44}]

a(n) = sum(k=1, n, (k/gcd(n, k))^3); \\ Michel Marcus, Apr 17 2021

a(n) = Sum_{d|n} A053819(d).

A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

Original entry on oeis.org

1, 39, 269, 1304, 3249, 10491, 17149, 42176, 66069, 126711, 162381, 350776, 373489, 668811, 873981, 1353216, 1424769, 2576691, 2482957, 4236696, 4613081, 6332859, 6448509, 11345344, 10168625, 14566071, 16073721, 22362296, 20535537, 34085259, 28658941, 43331584

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A343497, A368743, A372928, A372929.
Cf. A001157, A008683.
Cf. A002117, A013664, A157289.

f[p_, e_] := p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); 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^3*sigma(d, 2));

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)^3.
a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3)/zeta(6) = 1.181564... (A157289). (End)

A372918 a(n) = Sum_{k=1..n} gcd(k^3,n^2).

Original entry on oeis.org

1, 5, 11, 26, 29, 55, 55, 148, 141, 145, 131, 286, 181, 275, 319, 680, 305, 705, 379, 754, 605, 655, 551, 1628, 1145, 905, 2367, 1430, 869, 1595, 991, 3408, 1441, 1525, 1595, 3666, 1405, 1895, 1991, 4292, 1721, 3025, 1891, 3406, 4089, 2755, 2255, 7480, 4501, 5725, 3355

Offset: 1

Seiichi Manyama, May 16 2024

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

Cf. A069097, A078430, A343497, A372882.

a[n_] := Sum[GCD[k^3, n^2], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

PARI
```
a(n) = sum(k=1, n, gcd(k^3, n^2));
```

Multiplicative with a(p^e) = p^e * (1 + ((p-1)/p) * Sum_{i=1..2*e} p^(floor(2*i/3))). - Amiram Eldar, May 24 2024

cp's OEIS Frontend

A343513 a(n) = Sum_{k=1..n} (k/gcd(n, k))^3.

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A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

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A372918 a(n) = Sum_{k=1..n} gcd(k^3,n^2).

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