A327272 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.

1, 5, 42, 425, 17050, 54600, 11468100

Offset: 1

Author

Christopher J. Smyth, Sep 09 2019

Keywords

• nonn
• more

Comments

An algorithm for generating a(n) is given in the Pinner and Smyth link, where more details about a(n) can be found.

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.

Examples

			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.
		

Links

• 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 0 <= n <= 6

Crossrefs

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.

Formula

a(n) = A327267(Product_{k=0..n} prime(2^k)) = A327267(A325782(n+1)).