A059479 Number of 3 X 3 matrices with elements from {0,...,n-1} such that the middle element of each of the eight lines of three (rows, columns and diagonals) is the square (mod n) of the difference of the end elements.

1, 4, 9, 64, 25, 36, 49, 256, 729, 100, 121, 576, 169, 196, 225, 4096, 289, 2916, 361, 1600, 441, 484, 529, 2304, 15625, 676, 6561, 3136, 841, 900, 961, 16384, 1089, 1156, 1225, 46656, 1369, 1444, 1521, 6400, 1681, 1764, 1849, 7744, 18225, 2116, 2209

Offset: 1

Author

John W. Layman, Feb 15 2001

Keywords

• nonn
• mult

Comments

This sequence is multiplicative. - Mitch Harris, Apr 19 2005

The sequence enumerates the solutions of a system of polynomials equations modulo n, hence is multiplicative by the Chinese Remainder Theorem. The middle entry of the 3 X 3 is zero modulo n. - Michael Somos, Apr 30 2005

Links

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

Crossrefs

Cf. A008833, A013664, A082020.

Programs

Mathematica

f[p_, e_] := p^(3*e - (Mod[e, 2])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

PARI

a(n)=if(n<1,0,n^3/core(n)) /* Michael Somos, Apr 30 2005 */

Formula

a(n) = A008833(n)*n^2, where A008833(n) is the largest square that divides n.

Multiplicative with a(p^e) = p^(3e - (e % 2)). - Mitch Harris, Jun 09 2005

Dirichlet g.f.: zeta(s-2)*zeta(2s-6)/zeta(2s-4). - R. J. Mathar, Mar 30 2011

Sum_{k=1..n} a(k) ~ zeta(3/2) * n^(7/2) / (7*zeta(3)). - Vaclav Kotesovec, Sep 16 2020

Sum_{n>=1} 1/a(n) = 15*zeta(6)/Pi^2 = A082020 * A013664 = 1.546176... . - Amiram Eldar, Nov 03 2022