A327273 -id:A327273 - cp's OEIS frontend

A327271 Smallest modulus of any n X n integer determinant whose top row has all 1's and whose rows are pairwise orthogonal.

1, 2, 6, 8, 40, 48, 336, 128, 864, 1280, 8448, 3072, 39936

Offset: 1

Christopher J. Smyth, Sep 02 2019

a(n) = A327267(2^n), since 2^n = (p_1)^n is the Heinz code for the multiset {1,1,...,1}.
See Pinner and Smyth link below for more details, including an algorithm for computing A327267(n). Also, see file link below for {(n, a(n), matrix(n)), n <= 13}, where matrix(n) has minimal modulus determinant equal to a(n) among n X n matrices with top row all 1's and all rows orthogonal.
For the first 13 terms, the number of prime factors counted with multiplicity equals n-1: A001222(a(n))=n-1. How far does this hold? - Jon Maiga, Sep 07 2019

			a(3) = 6 because the matrix [[1,1,1],[1,-1,0],[1,1,-2]] has top row of 3 1's and all rows orthogonal, and minimal positive determinant equal to 6.

Chris Pinner and Chris Smyth, Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector
Christopher J. Smyth, List of n, a(n) and associated matrix for n up to 13

Subsequence of A327267, see comments; A327273 is similar, but determinant's top row is 1,2,2^2,...,2^{n-1}.

cp's OEIS Frontend

A327271 Smallest modulus of any n X n integer determinant whose top row has all 1's and whose rows are pairwise orthogonal.

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