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A322175
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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.
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0
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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
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OFFSET
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0,3
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COMMENTS
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a(n) <> 0 for 0 <= n <= 7, but a(n) = 0 for n >= 8.
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REFERENCES
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J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 694 pp. 90, 297, Ellipses Paris 2004.
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LINKS
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EXAMPLE
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For n = 2,
[ mu(1) mu(1) ] [ 1 1 ]
the matrix is [ ] = [ ]
[ mu(1) mu(2) ] [ 1 -1 ]
so a(2) = -2.
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = matdet(matrix(n, n, i, j, moebius(gcd(i, j)))); \\ Michel Marcus, Dec 03 2018
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CROSSREFS
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Cf. A008683, A001088 (determinant of n X n matrix M with M(i,j) = gcd(i,j))
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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