A118679 Absolute value of numerator of determinant of n X n matrix with M(i,j) = i/(i+1) if i=j otherwise 1.

1, 2, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 13, 47, 1, 229, 251, 137, 149, 1, 349, 47, 101, 433, 463, 1, 263, 43, 593, 157, 83, 701, 739, 389, 409, 859, 53, 59, 1, 1033, 83, 563, 587, 1223, 67, 331, 1, 1429, 1483, 769, 797, 127, 1709, 1, 457, 1889

Offset: 1

Author

Alexander Adamchuk, May 19 2006, Feb 03 2007

Keywords

• frac
• nonn

Comments

Numbers n such that a(n) = 1 are listed in A127852.

All a(n)>1 are prime belonging to A038889 (i.e., 17 is a square mod a(n)).

Crossrefs

Cf. A038889.

Cf. A118680, A127852, A127853.

Programs

Mathematica

Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ], {n, 1, 70} ]]
Table[ Numerator[ (n^2+3n-2)/(2(n+1)!) ], {n,1,100} ]

Formula

det(M) = (-1)^(n+1)*(n^2+3*n-2)/(2*(n+1)!), implying that a(n)=p, where p=A006530(n^2+3*n-2) is the largest prime divisor of (n^2+3*n-2), if p>n+1 or p=sqrt((n^2+3*n-2)/2); otherwise a(n)=1.

a(n) = Numerator[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ]].

a(n) = Numerator[ (n^2+3n-2)/(2(n+1)!) ] = Numerator[ ((2n+3)^2-17)/(4(n+1)!) ].

Extensions

Edited by Max Alekseyev, Jun 02 2009