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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.

%I A322175 #48 Dec 06 2018 11:29:25
%S A322175 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
%N 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.
%C A322175 a(n) <> 0 for 0 <= n <= 7, but a(n) = 0 for n >= 8.
%D A322175 J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 694 pp. 90, 297, Ellipses Paris 2004.
%e A322175 For n = 2,
%e A322175                [ mu(1)  mu(1) ]     [ 1  1 ]
%e A322175 the matrix is  [              ]  =  [      ]
%e A322175                [ mu(1)  mu(2) ]     [ 1 -1 ]
%e A322175 so a(2) = -2.
%t A322175 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 *)
%o A322175 (PARI) a(n) = matdet(matrix(n, n, i, j, moebius(gcd(i,j)))); \\ _Michel Marcus_, Dec 03 2018
%Y A322175 Cf. A008683, A001088 (determinant of n X n matrix M with M(i,j) = gcd(i,j))
%K A322175 sign
%O A322175 0,3
%A A322175 _Bernard Schott_, Dec 02 2018