A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

1, 1, 2, 8, 48, 288, 2880, 34560, 276480, 4423680, 79626240, 955514880, 21021327360, 420426547200, 7567677849600, 211894979788800, 6356849393664000, 127136987873280000, 3051287708958720000, 109846357522513920000, 2636312580540334080000, 105452503221613363200000

Offset

0,3

Comments

It appears, but has not been proved, that the ratios a(n+1)/a(n) give phi(2n+1) (A037225).

See A001088, A059381, and A059382 for determinants of matrices M defined by M(i,j) = gcd(i,j), gcd(i^2,j^2), and gcd(i^3,j^3), respectively.

Links

Table of n, a(n) for n=0..21.

Zhongshan Li, The determinants of GCD matrices, Linear Algebra Appl. 134 (1990), 137-143.

Formula

a(n) = Product_{k=1..n} phi(2*k-1), confirming the empirical observation above. This is because the set {1, 3, ..., 2n-1} is factor-closed. See the Li link. - Sela Fried, Feb 05 2026

Maple

A177066 := proc(n) M := Matrix(n) ; for i from 1 to n do for j from 1 to n do M[i, j] := igcd(2*i-1, 2*j-1) ; end do: end do: LinearAlgebra[Determinant](M) ; end proc: # R. J. Mathar, Dec 10 2010

Mathematica

a[n_]:=Det[Table[GCD[2i-1, 2j-1], {i, n}, {j, n}]]; Join[{1}, Array[a, 21]] (* Stefano Spezia, Feb 06 2026 *)

Prog

(PARI) a(n) = matdet(matrix(n, n, i, j, gcd(2*i-1, 2*j-1))); \\ Michel Marcus, Feb 06 2026

Crossrefs

Cf. A000010, A001088, A037225, A059381, A059382.

Sequence in context: A192251 A104190 A152661 * A356429 A391455 A228568

Adjacent sequences: A177063 A177064 A177065 * A177067 A177068 A177069

Keyword

nonn,easy

Author

John W. Layman, Dec 09 2010

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 05 2026

Status

approved