A332654 -id:A332654 - cp's OEIS frontend

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

[&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^n.

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

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

Original entry on oeis.org

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).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

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Magma

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

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

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