%I #37 Oct 30 2022 18:19:59
%S 1,4,18,144,900,16200,132300,2116800,28576800,714420000,8644482000,
%T 311201352000,4382752374000,143169910884000,4026653743612500,
%U 128852919795600000,2327405863808025000,125679916645633350000
%N Numerator of 1/det(M) where M is the n X n matrix with M[i,j] = 1/lcm(i,j).
%C The value of 1/det(M) is not always an integer! For example, 1/det(35) = 5029296746186844716050163189085401314000634765625/2. - _Harry J. Smith_, Jul 13 2009
%C 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
%C Values of n at which a(n) = a(n+1): 63, 127, 255, ..., . - _Robert G. Wilson v_, Aug 03 2015
%H Robert G. Wilson v, <a href="/A060841/b060841.txt">Table of n, a(n) for n = 1..400</a>
%F a(n) = (n!)^2 / (phi(1)*phi(2)*...*phi(n)) = (n!)^2 / A001088(n).
%e a(2) = 4 because the matrix M is [1,1/2; 1/2,1/2] and det(M) = 1/4.
%t d[n_] := Denominator[ Det[ Table[ GCD[1/i, 1/j], {i, n}, {j, n}]]; Array[d, 18]] (* _Robert G. Wilson v_, Aug 02 2015 *)
%o (PARI) vector(20, n, numerator(1/matdet(matrix(n, n, i, j, 1/lcm(i,j))))) \\ _Michel Marcus_, Aug 03 2015
%Y Cf. A000010, A001088, A060238, A260502, A260897.
%K nonn,easy
%O 1,2
%A Noam Katz (noamkj(AT)hotmail.com), May 02 2001
%E More terms from _Reiner Martin_, May 17 2001