A350933 - OEIS

%I #27 Feb 12 2024 10:41:13

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

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

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

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A350932%2B3.py">A350932+3.py</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>

%e a(2) = 19:

%e     5    2

%e     3    5

%e a(3) = 1115:

%e    11    2    5

%e     7   11    2

%e     3    7   11

%t 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 *)

%o (Python)

%o from itertools import permutations

%o from sympy import Matrix, prime

%o 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

%o (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

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

%K nonn,more

%O 0,2

%A _Stefano Spezia_, Jan 25 2022

%E a(5) from _Alois P. Heinz_, Jan 25 2022

%E a(6)-a(7) from _Lucas A. Brown_, Aug 27 2022