A187748 Determinant of the n X n matrix m_(i,j) = gcd(2^i-1, 2^j-1).

1, 2, 12, 144, 4320, 233280, 29393280, 7054387200, 3555411148800, 3519857037312000, 7201627498340352000, 28950542543328215040000, 237104943429858081177600000, 3853903750508913251460710400000, 126138269754156730720309051392000000, 8234306249551351381421774874869760000000, 1079270520128695625562952032849179443200000000, 282311265573183686952254740944556962034483200000000

Offset: 1

Author

Benoit Cloitre, Jan 03 2013

Keywords

• nonn

Crossrefs

Cf. A001088, A027375.

Programs

Mathematica

b[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#& ]; a[n_] := a[n] = If[n == 1, 1, a[n-1]*b[n]]; Array[a, 18] (* Jean-François Alcover, Dec 18 2015 *)
Table[Det[Table[GCD[2^i-1,2^j-1],{i,n},{j,n}]],{n,20}] (* Harvey P. Dale, Sep 23 2022 *)

PARI

a(n)=if(n<1,0,(1/2)*prod(k=1,n,sumdiv(k,d,moebius(d)*2^(k/d))))

Formula

a(n+1)/a(n) = A027375(n+1).

a(n) = (1/2)*Product_{k=1..n} Sum_{d|k} moebius(d)*2^(k/d).

a(n) ~ c * 2^(n*(n+1)/2), where c = 0.09412540696949274854160062245002977344042957885767746756023904566838799439... - Vaclav Kotesovec, Apr 19 2024