A228615 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as 2*(i + j) - 1 and 2*(i + j) + 1 are twin primes or not.

1, -1, -1, -1, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, -1, -1, 1, 1, 1, -1, 2, 8, -18, -9, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4096, -4096, 64, -20, -125, 5, -6, -216, 24, 152, 54872, -106742, 14045, 125, -21125, -274625, -274625, 10985, -16731, -970299, 1804275, 1312200, 373248, -691488, -192080

Offset

1,21

Comments

Conjecture: a(n) is nonzero for any n > 35.

Clearly this conjecture implies the twin prime conjecture.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..500 (* First 200 terms from Zhi-Wei Sun *)

Example

a(1) = 1 since 2*(1 + 1) - 1 = 3 and 2*(1 + 1) + 1 =5 are twin primes.

Mathematica

a[n_]:=a[n]=Det[Table[If[PrimeQ[2(i+j)-1]&&PrimeQ[2(i+j)+1], 1, 0], {i, 1, n}, {j, 1, n}]]
Table[a[n], {n, 1, 20}]
Table[Det[Table[If[AllTrue[2(i+j)+{1, -1}, PrimeQ], 1, 0], {i, k}, {j, k}]], {k, 60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2019 *)

Crossrefs

Cf. A001359, A006512, A069191, A228557, A228591.

Sequence in context: A268068 A073601 A051248 * A267823 A063664 A094147

Adjacent sequences: A228612 A228613 A228614 * A228616 A228617 A228618

Keyword

sign

Author

Zhi-Wei Sun, Aug 27 2013

Status

approved