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A069191
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Determinant of n X n matrix defined by m(i,j)=1 if i+j is a prime, m(i,j)=0 otherwise.
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16
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1, -1, -1, 0, 1, -1, -4, 4, 0, 0, 0, 0, 0, 0, 0, 4, 4, -9, -25, 121, 64, -576, -2304, 3600, 3136, -256, -144, 961, 24025, -47089, -345744, 1317904, 107584, -26896, -30976, 17424, 30976, -156025, -76729, 485809, 478864, -36481, -837225, 5776, 517198564, -15791440896, -16404230241, 45746793225
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OFFSET
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1,7
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COMMENTS
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Abs(a(n)) is always a perfect square.
A general result for Hankel determinants: Given any sequence a(0),a(1),... of numbers, let the n X n Hankel matrix A[i,j]=x if i=j=0, 0 if i+j even and a(((i+j)-1)/2) otherwise where 0<=i,j<n and the [n/2]X[n/2] Hankel matrix B[i,j]=a(i+j) if n even, a(i+j+1) if n odd. Then det(A) = (-1)^[n/2] det(B)^2 f where f=1 if n even, x if n odd. The proof uses Jacobi's determinant identity. - Michael Somos
Conjecture: Let M(n) be the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is prime or not. For each n = 24, 25, ..., the characteristic polynomial of M(n) is irreducible over the field of rational numbers, and the n eigenvalues can be listed as lambda(1), ..., lambda(n) such that lambda(1) > -lambda(2) > lambda(3) > -lambda(4) > .... > (-1)^n*lambda(n) > 0. - Zhi-Wei Sun, Aug 25 2013
That (-1)^(n(n-1)/2)*a(n) is a perfect square is a special case of a general theorem mentioned in A228591. - Zhi-Wei Sun, Aug 25 2013
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LINKS
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MATHEMATICA
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f[n_] := Det[ Table[ If[ PrimeQ[i + j], 1, 0], {i, 1, n}, {j, 1, n}]]; Table[ f[n], {n, 1, 45}]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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