A322175 Determinant of the symmetric n X n matrix M defined by M(i,j) = mu(gcd(i,j)) for 1 <= i,j <= n where mu is the Moebius function.

1, 1, -2, 4, 4, -8, -32, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Bernard Schott, Dec 02 2018

Keywords

• sign

Comments

a(n) <> 0 for 0 <= n <= 7, but a(n) = 0 for n >= 8.

Examples

			For n = 2,
               [ mu(1)  mu(1) ]     [ 1  1 ]
the matrix is  [              ]  =  [      ]
               [ mu(1)  mu(2) ]     [ 1 -1 ]
so a(2) = -2.
		

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 694 pp. 90, 297, Ellipses Paris 2004.

Crossrefs

Cf. A008683, A001088 (determinant of n X n matrix M with M(i,j) = gcd(i,j))

Programs

Mathematica

m[i_,j_] := MoebiusMu[GCD[i,j]]; a[n_] := Det[Table[m[i,j], {i,1,n}, {j,1,n}]]; Array[a, 30] (* Amiram Eldar, Dec 02 2018 *)

PARI

a(n) = matdet(matrix(n, n, i, j, moebius(gcd(i,j)))); \\ Michel Marcus, Dec 03 2018