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%I #18 Jun 12 2022 06:42:57
%S 1,2,-5,-70,1275,97748,-2713585,-251983958,9651414311,1137214908700,
%T -268100912462097,-16553358418854560,4303513869962179379,
%U 602501593820064477686,-50199332236439321779977,-7847812115804566640572424,2754406130856424049914030863
%N Determinant of n X n matrix whose rows are cyclic permutations of 2..prime(n).
%H Alois P. Heinz, <a href="/A066933/b066933.txt">Table of n, a(n) for n = 0..300</a>
%F 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
%e a(3) = -70 because this is the determinant of [(2,3,5), (3,5,2), (5,2,3)].
%p a:= n-> LinearAlgebra[Determinant](Matrix(n,
%p (i, j)-> ithprime(1+irem(i+j-2, n)))):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Dec 09 2016
%t 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} ]
%o (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
%Y Cf. A052182.
%K easy,sign
%O 0,2
%A _Robert G. Wilson v_, Jan 24 2002
%E a(0)=1 prepended by _Alois P. Heinz_, Dec 09 2016