A066933 Determinant of n X n matrix whose rows are cyclic permutations of 2..prime(n).

1, 2, -5, -70, 1275, 97748, -2713585, -251983958, 9651414311, 1137214908700, -268100912462097, -16553358418854560, 4303513869962179379, 602501593820064477686, -50199332236439321779977, -7847812115804566640572424, 2754406130856424049914030863

Offset: 0

Author

Robert G. Wilson v, Jan 24 2002

Keywords

• easy
• sign

Examples

			a(3) = -70 because this is the determinant of [(2,3,5), (3,5,2), (5,2,3)].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A052182.

Programs

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(n,
       (i, j)-> ithprime(1+irem(i+j-2, n)))):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 09 2016

Mathematica

f[ n_ ] := Module[ {a = Table[ Prime[ i ], {i, 1, n} ], m = {}, k = 0}, While[ k < n, m = Append[ m, RotateLeft[ a, k ] ]; k++ ]; Det[ m ] ]; Table[ f[ n ], {n, 1, 16} ]

PARI

a(n) = matdet(matrix(n, n, i, j, prime(1+lift(Mod(i+j-2, n))))); \\ Michel Marcus, Aug 11 2019; corrected Jun 12 2022

Formula

Conjecture: a(n) = (-1)^(n+floor(n/2))*Res(f(n) , x^n - 1), where Res is the resultant, and f(n)=Sum_{k=1..n} prime(k)*x^k. - Benedict W. J. Irwin, Dec 07 2016

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 09 2016