A367133 Rank of the n X n matrix whose terms are the n^2 values of isprime(x) from 1 to n^2.

0, 2, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 11, 7, 9, 9, 16, 7, 19, 9, 13, 11, 23, 9, 20, 13, 19, 13, 29, 9, 31, 17, 21, 17, 25, 13, 37, 19, 25, 17, 41, 13, 43, 21, 25, 23, 47, 17, 43, 21, 33, 25, 53, 19, 41, 25, 37, 29, 59, 17, 61, 31, 37, 33, 49, 21, 67, 33, 45, 25

Offset

1,2

Comments

Traces of these matrices are A221490.

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 largest square submatrix with a nonzero determinant within this matrix is of dimension 3. Therefore, the rank is 3, and so a(4)=3.

Maple

f:= proc(n) local i; LinearAlgebra:-Rank(Matrix(n, n, [seq(`if`(isprime(i), 1, 0), i=1..n^2)])) end proc:
map(f, [$1..100]); # Robert Israel, Nov 11 2023

Mathematica

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

Prog

(PARI) a(n) = matrank(matrix(n, n, i, j, isprime((i-1)*n+j))); \\ Michel Marcus, Nov 07 2023
(Python)
from sympy import Matrix, isprime
def A367133(n): return Matrix(n, n, [int(isprime(i)) for i in range(1, n**2+1)]).rank() # Chai Wah Wu, Nov 16 2023

Crossrefs

Cf. A010051, A064866, A289777, A367077, A221490.

Sequence in context: A064917 A154727 A323075 * A065070 A070800 A138663

Adjacent sequences: A367130 A367131 A367132 * A367134 A367135 A367136

Keyword

nonn

Author

Andres Cicuttin, Nov 06 2023

Status

approved