A367077 - OEIS

%I #34 Nov 16 2023 12:21:57

%S 0,-1,-1,0,1,0,-2,0,0,0,-1,0,0,0,0,0,0,0,-5,0,0,0,-15,0,0,0,0,0,400,0,

%T -196,0,0,0,0,0,4224,0,0,0,-44304,0,-537138,0,0,0,-4152330,0,0,0,0,0,

%U -59171526,0,0,0,0,0,-1681340912,0,330218571840,0,0,0,0,0,-349982854480,0,0,0

%N Determinant of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

%C Traces of these matrices are A221490.

%C Consider the sequence b(n) defined as 0 when a(n) is 0 and 1 otherwise. What is the value of the limit as n approaches infinity of Sum_{j<=n} b(j)/n provided that this limit exists?

%e For n=4, we consider the first n^2=16 terms of the characteristic function of primes (A010051): (0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0). These terms form a matrix by arranging them in 4 consecutive subsequences of 4 terms each:

%e   0, 1, 1, 0;

%e   1, 0, 1, 0;

%e   0, 0, 1, 0;

%e   1, 0, 0, 0;

%e and the determinant of this matrix is zero, so a(4)=0.

%t mat[n_,i_,j_]:=Boole[PrimeQ[(i-1)*n+j]];

%t a[n_]:=Det@Table[mat[n,i,j],{i,1,n},{j,1,n}];

%t Table[a[n],{n,1,70}]

%o (PARI) a(n) = matdet(matrix(n, n, i, j, isprime((i-1)*n+j))); \\ _Michel Marcus_, Nov 07 2023

%o (Python)

%o from sympy import Matrix, isprime

%o def A367077(n): return Matrix(n,n,[int(isprime(i)) for i in range(1,n**2+1)]).det() # _Chai Wah Wu_, Nov 16 2023

%Y Cf. A010051, A064866, A289777, A367133, A221490.

%K sign

%O 1,7

%A _Andres Cicuttin_, Nov 05 2023