A373582 a(n) = Sum_{k=1..n} Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k] * f(x,y,z) * A023900(k), where f(x,y,z) = x^2 + y^2 - z^2.

1, 0, -117, 0, -350, 0, -4263, 0, -7533, 0, -27225, 0, -17914, 0, 62100, 0, -53176, 0, -250173, 0, 83790, 0, -541167, 0, -168750, 0, -557685, 0, -459186, 0, -1801875, 0, 533610, 0, 2249100, 0, -1223886, 0, 3157596, 0, -1849100, 0, -6717417, 0, 3863700, 0, -9602523

Offset

1,3

Comments

Sign(a(n)) appears to be equal to A318657(n).

For n > 1, mod(a(n),2) appears to be equal to A354033(n).

Links

Table of n, a(n) for n=1..47.

Formula

a(n) = Sum_{k=1..n} Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k] * f(x,y,z) * A023900(k), where f(x,y,z) = x^2 + y^2 - z^2.

Mathematica

nn = 47; a[n_] = DivisorSum[n, MoebiusMu[#]  # &]; p = 2; f = x^p + y^p - z^p; ParallelTable[Sum[Sum[Sum[Sum[If[GCD[f, n] == k, f, 0] a[k], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 1, nn}]

Crossrefs

Cf. A368197, A023900, A318657, A354033.

Sequence in context: A009617 A201728 A009687 * A050660 A176568 A023682

Adjacent sequences: A373579 A373580 A373581 * A373583 A373584 A373585

Keyword

sign

Author

Mats Granvik, Jun 10 2024

Status

approved