A369947 - OEIS

%I #17 Feb 12 2024 12:34:33

%S 1,2,11,286,86087,9603283,1764195984

%N a(n) is the maximal 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) = 11:

%e   3, 2;

%e   2, 5.

%e a(3) = 286:

%e    3, 11, 5;

%e   11,  5, 7;

%e    5,  7, 2.

%e a(4) = 86087:

%e    7,  3, 13, 17;

%e    3, 13, 17,  2;

%e   13, 17,  2, 11;

%e   17,  2, 11,  5.

%t a[n_] := Max[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 A369947(n): return max(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, A368352.

%Y Cf. A369946 (minimal), A350933 (maximal absolute value), A369949, A350940 (maximal permanent).

%K nonn,hard,more

%O 0,2

%A _Stefano Spezia_, Feb 06 2024

%E a(6) from _Michel Marcus_, Feb 08 2024