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

1,2

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.

Links

Table of n, a(n) for n=1..7.

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

Formula

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

Example

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.

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.

Sequence in context: A126765 A228793 A330628 * A024492 A217805 A217808

Adjacent sequences: A327269 A327270 A327271 * A327273 A327274 A327275

Keyword

nonn,more

Author

Christopher J. Smyth, Sep 09 2019

Status

approved