A327271 - OEIS

%I #31 Nov 28 2019 07:53:14

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

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

%C a(n) = A327267(2^n), since 2^n = (p_1)^n is the Heinz code for the multiset {1,1,...,1}.

%C 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.

%C 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

%H Chris Pinner and Chris Smyth, <a href="https://www.maths.ed.ac.uk/~chris/papers/MinimalLattices040919.pdf">Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector</a>

%H Christopher J. Smyth, <a href="/A327271/a327271.pdf">List of n, a(n) and associated matrix for n up to 13</a>

%e 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.

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

%K nonn,more

%O 1,2

%A _Christopher J. Smyth_, Sep 02 2019