A060841 - OEIS

A060841

Numerator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).

1, 4, 18, 144, 900, 16200, 132300, 2116800, 28576800, 714420000, 8644482000, 311201352000, 4382752374000, 143169910884000, 4026653743612500, 128852919795600000, 2327405863808025000, 125679916645633350000

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OFFSET

1,2

COMMENTS

The value of 1/det(M) is not always an integer! For example, 1/det(35) = 5029296746186844716050163189085401314000634765625/2. - Harry J. Smith, Jul 13 2009

Conjecture: 1/det(M) is an integer only for n: 1 - 34, 36 and 38. All denominators are powers of two (A000079). But not all powers of two are present. See A260502. - Robert G. Wilson v, Aug 02 2015

Values of n at which a(n) = a(n+1): 63, 127, 255, ..., . - Robert G. Wilson v, Aug 03 2015

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..400

FORMULA

a(n) = (n!)^2 / (phi(1)*phi(2)*...*phi(n)) = (n!)^2 / A001088(n).

EXAMPLE

a(2) = 4 because the matrix M is [1,1/2; 1/2,1/2] and det(M) = 1/4.

MATHEMATICA

d[n_] := Denominator[ Det[ Table[ GCD[1/i, 1/j], {i, n}, {j, n}]]; Array[d, 18]] (* Robert G. Wilson v, Aug 02 2015 *)

PROG

(PARI) vector(20, n, numerator(1/matdet(matrix(n, n, i, j, 1/lcm(i, j))))) \\ Michel Marcus, Aug 03 2015

CROSSREFS