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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.
0
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
0,3
COMMENTS
a(n) <> 0 for 0 <= n <= 7, but a(n) = 0 for n >= 8.
REFERENCES
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 694 pp. 90, 297, Ellipses Paris 2004.
EXAMPLE
For n = 2,
[ mu(1) mu(1) ] [ 1 1 ]
the matrix is [ ] = [ ]
[ mu(1) mu(2) ] [ 1 -1 ]
so a(2) = -2.
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 *)
PROG
(PARI) a(n) = matdet(matrix(n, n, i, j, moebius(gcd(i, j)))); \\ Michel Marcus, Dec 03 2018
CROSSREFS
Cf. A008683, A001088 (determinant of n X n matrix M with M(i,j) = gcd(i,j))
Sequence in context: A264190 A006967 A296229 * A298117 A122033 A281122
KEYWORD
sign
AUTHOR
Bernard Schott, Dec 02 2018
STATUS
approved