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A228623 Determinant of the n X n matrix with (i,j)-entry (i,j = 0,...,n-1) equal to 1 or 0 according as n + i - j and n - i + j are both prime or not.

%I A228623 #9 Aug 28 2013 03:03:05
%S A228623 0,1,1,1,0,-1,0,0,0,-4,-1,0,0,0,-6,0,0,-144,0,0,0,-1,168,1024,420,0,0,
%T A228623 0,-1,-9801,0,144,0,0,3072,7056,0,0,-42346434,0,0,-331776,0,0,
%U A228623 36528128,-104976,96545145,0,34665386,-62500,2826240,2025,0,-23174596,0,0,255578880,-4,-3,990172089
%N A228623 Determinant of the n X n matrix with (i,j)-entry (i,j = 0,...,n-1) equal to 1 or 0 according as n + i - j and n - i + j are both prime or not.
%C A228623 Conjecture: a(n) is nonzero if n is odd and greater than 120.
%C A228623 This implies Goldbach's conjecture for even numbers of the form 4*k + 2.
%H A228623 Zhi-Wei Sun, <a href="/A228623/b228623.txt">Table of n, a(n) for n = 1..400</a>
%e A228623 a(1) = 0 since 1 + 0 - 0 = 1 is not a prime.
%t A228623 a[n_]:=Det[Table[If[PrimeQ[n+j-i]==True&&PrimeQ[n+i-j]==True,1,0],{i,0,n-1},{j,0,n-1}]]
%t A228623 Table[a[n],{n,1,20}]
%Y A228623 Cf. A002372, A228591, A228615, A228616, A228557, A228559.
%K A228623 sign
%O A228623 1,10
%A A228623 _Zhi-Wei Sun_, Aug 27 2013