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

1,7

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?

Links

Table of n, a(n) for n=1..70.

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A255309 A335446 A335460 * A368073 A086071 A322212

Adjacent sequences: A367074 A367075 A367076 * A367078 A367079 A367080

Keyword

sign

Author

Andres Cicuttin, Nov 05 2023

Status

approved