|
%I #18 Sep 12 2019 18:27:11
%S 1,5,42,425,17050,54600,11468100
%N Smallest modulus of any (n+1) X (n+1) integer determinant whose top row is 1,2,2^2,...,2^n and whose rows are pairwise orthogonal.
%C An algorithm for generating a(n) is given in the Pinner and Smyth link, where more details about a(n) can be found.
%C Also, see file link below for {(n,a(n),matrix(n)),0 <= n <= 6}, where matrix(n) has minimal modulus determinant equal to a(n) among (n+1) X (n+1) matrices with top row 1,2,2^2,...,2^n and all rows orthogonal.
%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="/A327272/a327272.pdf">List of n, a(n) and associated matrix for 0 <= n <= 6</a>
%F a(n) = A327267(Product_{k=0..n} prime(2^k)) = A327267(A325782(n+1)).
%e a(2) =42 since det([[1,2,4],[2,-3,1],[2,1,-1]]) = 42 and is the smallest positive determinant with top row [1,2,2^2] and all entries integers, and rows orthogonal.
%Y Subsequence of A327267-- see comments; A327269 is similar, but determinant's top row is 1,2,...,n; A327271 is similar, but determinant's top row consists of n 1's.
%K nonn,more
%O 1,2
%A _Christopher J. Smyth_, Sep 09 2019
|
