A367077 Determinant of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

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, -196, 0, 0, 0, 0, 0, 4224, 0, 0, 0, -44304, 0, -537138, 0, 0, 0, -4152330, 0, 0, 0, 0, 0, -59171526, 0, 0, 0, 0, 0, -1681340912, 0, 330218571840, 0, 0, 0, 0, 0, -349982854480, 0, 0, 0

Offset: 1

Author

Andres Cicuttin, Nov 05 2023

Keywords

• sign

Comments

Traces of these matrices are A221490.

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?

Examples

			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:
  0, 1, 1, 0;
  1, 0, 1, 0;
  0, 0, 1, 0;
  1, 0, 0, 0;
and the determinant of this matrix is zero, so a(4)=0.
		

Crossrefs

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

Programs

Mathematica

mat[n_,i_,j_]:=Boole[PrimeQ[(i-1)*n+j]];
a[n_]:=Det@Table[mat[n,i,j],{i,1,n},{j,1,n}];
Table[a[n],{n,1,70}]

PARI

a(n) = matdet(matrix(n, n, i, j, isprime((i-1)*n+j))); \\ Michel Marcus, Nov 07 2023

Python

from sympy import Matrix, isprime
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