A366561 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^2 - y^2.

1, 2, 2, 4, 0, 5, 8, 0, 0, 8, 16, 0, 0, 0, 9, 8, 8, 10, 0, 0, 10, 36, 0, 0, 0, 0, 0, 13, 32, 0, 0, 8, 0, 0, 0, 24, 36, 0, 24, 0, 0, 0, 0, 0, 21, 32, 32, 0, 0, 18, 0, 0, 0, 0, 18, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40

Offset

1,2

Comments

Row n appears to have sum n^2. The number of nonzero terms in row n is A366563(n). Sum_{k=1..n} T(n,k)*A023900(k)/n = A366562(n).

Links

Table of n, a(n) for n=1..78.

Formula

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^2 - y^2.

Conjecture 1: T(n,n) = A062803(n).

Conjecture 2: T(n,1) = A082953(n).

Example

{
{1}, = 1^2
{2, 2}, = 2^2
{4, 0, 5}, = 3^2
{8, 0, 0, 8}, = 4^2
{16, 0, 0, 0, 9}, = 5^2
{8, 8, 10, 0, 0, 10}, = 6^2
{36, 0, 0, 0, 0, 0, 13}, = 7^2
{32, 0, 0, 8, 0, 0, 0, 24}, = 8^2
{36, 0, 24, 0, 0, 0, 0, 0, 21}, = 9^2
{32, 32, 0, 0, 18, 0, 0, 0, 0, 18}, = 10^2
{100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21}, = 11^2
{32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40} = 12^2
}

Mathematica

nn = 12; f = x^2 - y^2; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

Prog

(PARI) T(n, k) = sum(x=1, n, sum(y=1, n, gcd(x^2 - y^2, n) == k)); \\ Michel Marcus, Oct 14 2023

Crossrefs

Cf. A127649, A000290, A062803, A082953.

Sequence in context: A246816 A127786 A030207 * A061006 A080736 A326127

Adjacent sequences: A366558 A366559 A366560 * A366562 A366563 A366564

Keyword

nonn,tabl

Author

Mats Granvik, Oct 13 2023

Status

approved