

A228574


Determinant of the 2*n X 2*n matrix with (i,j)entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod 4 or not.


6



0, 1, 0, 1, 0, 16, 0, 1, 0, 1, 0, 6561, 0, 0, 0, 0, 0, 0, 0, 6561, 0, 456976, 0, 65536, 0, 84934656, 0, 12745506816, 0, 335563778560000, 0, 1105346784523536, 0, 441194850625, 0, 986262467993856, 0, 80385880645971214336, 0, 6387622009837971841
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OFFSET

1,6


COMMENTS

For the (2*n1) X (2*n1) determinant with (i,j)entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod 4 or not, it is easy to see that it vanishes since sum_{i=1}^{2*n1} (i + tau(i)  1) is not a multiple of 4 for any permutation tau of {1,...,2n1}.
Conjecture: a(2*n1) = 0 for all n > 0, and a(2*n) is nonzero when n > 9.
ZhiWei Sun could prove the following related result:
Let m be any positive even integer, and let D(m, n) denote the n X n determinant with (i,j)entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod m or not. Then (1)^{n*(n1)/2}*D(m,n) is always an mth power. (It is easy to see that D(m,n) = 0 if m does not divide n^2.)


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..150


MATHEMATICA

a[n_]:=a[n]=Det[Table[If[Mod[i+j, 4]==1&&PrimeQ[i+j]==True, 1, 0], {i, 1, 2n}, {j, 1, 2n}]]
Table[a[n], {n, 1, 20}]


CROSSREFS

Cf. A002144, A069191, A228591, A228552, A228557, A228559, A228561, A228615, A228616.
Sequence in context: A037217 A109075 A187585 * A007791 A294699 A070570
Adjacent sequences: A228571 A228572 A228573 * A228575 A228576 A228577


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Aug 25 2013


STATUS

approved



