%I #19 Feb 12 2024 12:35:06
%S 1,2,-19,-1115,-57935,-5696488,-2307021183
%N a(n) is the minimal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>.
%e a(2) = -19:
%e 2, 5;
%e 5, 3.
%e a(3) = -1115:
%e 3, 7, 11;
%e 7, 11, 2;
%e 11, 2, 5.
%e a(4) = -57935:
%e 7, 5, 17, 2;
%e 5, 17, 2, 3;
%e 17, 2, 3, 11;
%e 2, 3, 11, 13.
%t 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]]
%o (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 (d<m, m = d)); m; \\ _Michel Marcus_, Feb 08 2024
%o (Python)
%o from itertools import permutations
%o from sympy import primerange, prime, Matrix
%o 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
%Y Cf. A024356, A368351.
%Y Cf. A369947 (maximal), A350933 (maximal absolute value), A369949, A350939 (minimal permanent).
%K sign,hard,more
%O 0,2
%A _Stefano Spezia_, Feb 06 2024
%E a(6) from _Michel Marcus_, Feb 08 2024