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a(n) is the minimal determinant of an n X n Hankel matrix using the first 2*n - 1 prime numbers.
4

%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