A369946 a(n) is the minimal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.

1, 2, -19, -1115, -57935, -5696488, -2307021183

Offset: 0

Author

Stefano Spezia, Feb 06 2024

Keywords

• sign
• hard
• more

Examples

			a(2) = -19:
  2, 5;
  5, 3.
a(3) = -1115:
   3,  7, 11;
   7, 11,  2;
  11,  2,  5.
a(4) = -57935:
   7,  5, 17,  2;
   5, 17,  2,  3;
  17,  2,  3, 11;
   2,  3, 11, 13.
		

Links

• Wikipedia, Hankel matrix.

Crossrefs

Cf. A024356, A368351.

Cf. A369947 (maximal), A350933 (maximal absolute value), A369949, A350939 (minimal permanent).

Programs

Mathematica

a[n_] := Min[Table[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]]

PARI

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

Python

from itertools import permutations
from sympy import primerange, prime, Matrix
def A369946(n): return min(Matrix([p[i:i+n] for i in range(n)]).det() for p in permutations(primerange(prime((n<<1)-1)+1))) if n else 1 # Chai Wah Wu, Feb 12 2024

Extensions

a(6) from Michel Marcus, Feb 08 2024