A228557 -id:A228557 - cp's OEIS frontend

A228591 Determinant of the n X n (0,1)-matrix with (i,j)-entry equal to 1 if and only if i + j is 2 or an odd composite number.

1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -9, 81, 9, -1225, -2500, 2500, 2500, -225, -121, 841, 19044, -29584, -355216, 1527696, 141376, -40000, -40000, 10000, 59536, -258064, -139876, 935089, 885481, -16384, -1876900, 1710864, 818875456, -22896531856, -23799232900, 66328911936, 158281561, -45320023225

Offset: 1

Zhi-Wei Sun, Aug 27 2013

sign

Conjecture: a(n) = 0 for no n > 15.
We observe that (-1)^{n*(n-1)/2}*a(n) is always a square. This is a special case of the following general result established by Zhi-Wei Sun.
Theorem: Let M = (m_{i,j}) be an n X n symmetric matrix over a commutative ring. Suppose that the (i,j)-entry m_{i,j} is zero whenever i + j is even and greater than 2. If n is even, then (-1)^{n/2}*det(M) = D(n)^2, where D(n) denotes the determinant |m_{2i,2j-1}|{i,j = 1,...,n/2}. If n is odd, then (-1)^{(n-1)/2}*det(M) = m{1,1}*D(n)^2, where D(n) is the determinant |m_{2i,2j+1}|_{i,j = 1,...,(n-1)/2}.
This theorem extends the result mentioned in A069191.

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Cf. A069191, A071524, A228552, A228557, A228559, A228561, A228574, A228578, A228615, A228616.

a[n_]:=a[n]=Det[Table[If[(i+j==2)||(Mod[i+j,2]==1&&PrimeQ[i+j]==False),1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,50}]

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.

Original entry on oeis.org

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

Zhi-Wei Sun, Aug 27 2013

sign

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

Clearly this conjecture implies the twin prime conjecture.

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

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

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

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 *)

A228559 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a Sophie Germain prime or not.

Original entry on oeis.org

1, -1, -1, 1, 0, -1, -1, 1, 1, -1, 0, 1, 1, -1, -4, 16, 0, -64, -64, 64, 0, 0, 0, 0, 0, -64, -64, 64, 0, -16, -4, 1, 1, -1, 0, 4, 16, -64, -144, 324, 0, -81, -9, 1, 4, -16, 0, 64, 64, -64, 0, 0, 0, 0, 0, 262144, 4194304, -67108864, 0, 1073741824

Offset: 1

Zhi-Wei Sun, Aug 25 2013

sign

If p > 3 and 2*p+1 are both prime, then p == -1 (mod 6). If tau is a permutation of {1,...,n}, and i + tau(i) is a Sophie Germain prime for each i = 1,...,n, then n*(n+1) = sum_{i=1}^n (i + tau(i)) is congruent to -n or 2 - (n - 1) or 3 - (n - 1) or 3 + 3 - (n - 2) modulo 6, which is impossible when n == -1 (mod 6). Therefore a(6*n-1) = 0 for all n > 0.
Note also that (-1)^{n*(n-1)/2}*a(n) is always a square in view of the comments in A228591.
Conjecture: a(n) is nonzero if n is greater than 55 and not congruent to 5 modulo 6.
Zhi-Wei Sun also had some similar conjectures. For example, if b(n) denotes the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j and 2*(i + j) - 1 are both prime or not, then b(n) is nonzero when n is greater than 125 and not congruent to 3 modulo 6. (Just like a(6*n-1) = 0, we can show that b(6*n-3) = 0.)

			a(1) = 1 since 1 + 1 is a Sophie Germain prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A005384, A228591, A069191, A071524, A228552, A228557, A228561, A228574, A228578, A228615, A228616, A228548, A228549.

a[n_]:=a[n]=Det[Table[If[PrimeQ[i+j]==True&&PrimeQ[2(i+j)+1]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,20}]
Det/@Table[If[AllTrue[{i+j,2(i+j)+1},PrimeQ],1,0],{n,60},{i,n},{j,n}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2017 *)

A228561 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.

Original entry on oeis.org

1, -1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0, 4, -16, -9, 25, 4, -81, -81, 81, 841, -5929, -3969, 19600, 69169, -667489, -285156, 80656, 276676, -790321, -60025, 3136, 10816, -40000, -45369, 221841, 86436, -168100, -12100, 13225, 11881, -87616, -71289, 729

Offset: 1

Zhi-Wei Sun, Aug 25 2013

sign

Conjecture: a(n) is nonzero for each n > 28.

This implies that there are infinitely many primes p with 4*p^2 + 1 also prime. Note also that (-1)^{n*(n-1)/2}*a(n) is always a square in view of the comments in A228591.

			a(1) = 1 since  1 + 1 = 2 and 4*2^2 + 1 = 17 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A052291, A069191, A071524, A228591, A228552, A228557, A228559, A228574, A228578, A228615, A228616.

a[n_]:=a[n]=Det[Table[If[PrimeQ[i+j]==True&&PrimeQ[4(i+j)^2+1]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]

A228574 Determinant of the 2n X 2n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod 4 or not.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Aug 25 2013

nonn

For the (2*n-1) X (2*n-1) 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*n-1} (i + tau(i) - 1) is not a multiple of 4 for any permutation tau of {1,...,2n-1}.
Conjecture: a(2*n-1) = 0 for all n > 0, and a(2*n) is nonzero when n > 9.
Zhi-Wei 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*(n-1)/2}*D(m,n) is always an m-th power. (It is easy to see that D(m,n) = 0 if m does not divide n^2.)

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Cf. A002144, A069191, A228591, A228552, A228557, A228559, A228561, A228615, A228616.

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}]

A071524 Determinant of n X n matrix defined by m(i,j)=1 if i^2+j^2 is a prime, m(i,j)=0 otherwise.

Original entry on oeis.org

1, -1, -1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 16, 25, -25, -100, 1, 81, -16, -36, 0, 1764, -3136, -196, 324, 16, -225, -1764, 1521, 9, -3969, -4356, 4761, 9, -1225, -19881, 5041, 156816, -312481, -167281, 219024, 3600, -186624, -158404, 5541316, 3020644, -19554084, -1350244, 198810000

Offset: 1

Benoit Cloitre, Jun 02 2002

Terms are also perfect squares.
Conjecture: a(n) = 0 for no n > 28. - Zhi-Wei Sun, Aug 26 2013
General conjecture: Let m be any nonnegative integer, and let a(m,n) be the n X n determinant with (i,j)-entry equal to 1 or 0 according as i^{2^m}+j^{2^m} is prime or not. Then a(m,n) is nonzero for large n. (It can be proved that (-1)^(n*(n-1)/2)*a(m,n) is always a square, see the comments in A228591.) - Zhi-Wei Sun, Aug 26-27 2013

Cf. A069191, A228591, A228552, A228557, A228559, A228561, A228574, A228578.

a[n_]:=a[n]=Det[Table[If[PrimeQ[i^2+j^2]==True, 1, 0], {i, 1, n}, {j, 1, n}]]; Table[a[n], {n, 1, 30}] (* Zhi-Wei Sun, Aug 26 2013 *)
Table[Det[Table[If[PrimeQ[a^2+b^2],1,0],{a,n},{b,n}]],{n,60}] (* Harvey P. Dale, May 31 2019 *)

for(n=1,60,print1(((matdet(matrix(n,n,i,j,isprime(i^2+j^2))))),","))

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.

Original entry on oeis.org

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, 0, -1, -9801, 0, 144, 0, 0, 3072, 7056, 0, 0, -42346434, 0, 0, -331776, 0, 0, 36528128, -104976, 96545145, 0, 34665386, -62500, 2826240, 2025, 0, -23174596, 0, 0, 255578880, -4, -3, 990172089

Offset: 1

Zhi-Wei Sun, Aug 27 2013

sign

Conjecture: a(n) is nonzero if n is odd and greater than 120.

This implies Goldbach's conjecture for even numbers of the form 4*k + 2.

			a(1) = 0 since 1 + 0 - 0 = 1 is not a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A002372, A228591, A228615, A228616, A228557, A228559.

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}]]
Table[a[n],{n,1,20}]

A228624 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not.

Original entry on oeis.org

0, 0, -1, 0, 1, 0, 0, 1, 1, 1, 0, -1, 1, 0, 0, -1, 2, 3, -3, -1, 0, 1, -1, -2, -5, 13, -7, -7, -6, 1, 8, -1, -17, 25, 13, -12, 11, 12, -11, -12, -4, 1, 1, -66, -60, -26, -13, 40, -67, -1, 82, 81, -49, -32, 68, 103, -222, 503, -39, -134

Offset: 1

Zhi-Wei Sun, Aug 28 2013

sign

Conjecture: a(n) is nonzero for any n > 21.
Zhi-Wei Sun also made the following similar conjecture:
  Let A(n) be the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a cube or not. Then A(n) is nonzero for any n > 176.

			a(1) = 0 since 1 + 1 = 2 is not a square.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A000290, A069191, A228591, A228557, A228559, A228615, A228616, A228623.

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Det[Table[If[SQ[i+j]==True,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,30}]

a(n)=matdet(matrix(n,n,i,j,issquare(i+j))) \\ Ralf Stephan, Sep 17 2013

cp's OEIS Frontend

A228591 Determinant of the n X n (0,1)-matrix with (i,j)-entry equal to 1 if and only if i + j is 2 or an odd composite number.

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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.

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A228559 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a Sophie Germain prime or not.

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A228561 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.

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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.

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A071524 Determinant of n X n matrix defined by m(i,j)=1 if i^2+j^2 is a prime, m(i,j)=0 otherwise.

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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.

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A228624 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a square or not.

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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.

A228574 Determinant of the 2n X 2n matrix with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod 4 or not.