A350933 Maximal determinant of an n X n Toeplitz matrix using the first 2*n - 1 prime numbers.

1, 2, 19, 1115, 86087, 9603283, 2307021183, 683793949387

Offset: 0

Author

Stefano Spezia, Jan 25 2022

Keywords

• nonn
• more

Comments

For n X n Hankel matrices the same maximal determinants appear.

Examples

			a(2) = 19:
    5    2
    3    5
a(3) = 1115:
   11    2    5
    7   11    2
    3    7   11
		

Links

• Lucas A. Brown, A350932+3.py
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A024356, A318173, A350931, A350932 (minimal), A369946, A369947.

Programs

Mathematica

a[n_] := Max[Table[Abs[Det[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n], {Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]] (* Stefano Spezia, Feb 06 2024 *)

PARI

a(n) = my(v=[1..2*n-1], m=-oo, d); forperm(v, p, d = abs(matdet(matrix(n, n, i, j, prime(p[i+j-1])))); if (d>m, m = d)); m; \\ Michel Marcus, Feb 08 2024

Python

from itertools import permutations
from sympy import Matrix, prime
def A350933(n): return max(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).det() for p in permutations(prime(i) for i in range(1,2*n))) # Chai Wah Wu, Jan 27 2022

Extensions

a(5) from Alois P. Heinz, Jan 25 2022

a(6)-a(7) from Lucas A. Brown, Aug 27 2022