A327267 Minimal determinant of a finite-dimensional integer lattice having an orthogonal basis containing a given vector with all entries positive.

0, 1, 2, 2, 3, 5, 4, 6, 4, 10, 5, 6, 6, 17, 13, 8, 7, 18, 8, 22, 10, 26, 9, 42, 6, 37, 12, 18, 10, 42, 11, 40, 29, 50, 25, 20, 12, 65, 20, 24, 13, 42, 14, 54, 34, 82, 15, 32, 8, 38, 53, 38, 16, 78, 34, 114, 34, 101, 17, 30, 18, 122, 12, 48, 15, 30

Offset: 1

Author

Christopher J. Smyth, Aug 31 2019

Keywords

• nonn

Comments

The given basis vector (k_1,...,k_r) is encoded as n = p_{k_1}...p_{k_r}, where p_j is the j-th prime (Heinz encoding). Then a(n) is the minimal (positive) determinant of all integer r X r matrices with top row (k_1,...,k_r) and all rows pairwise orthogonal.

The values of n and a(n) are independent of the order of the k_j's; they depend only on the multiset {k_1,...,k_r}.

An algorithm for computing a(n) is described in the Pinner and Smyth link below. It has been implemented in Maple. More properties of this sequence are also discussed in this paper.

Examples

			For n = 6 = p_1*p_2, the given basis vector is (1,2), and a(n)=5 because the matrix ((1,2),(-2,1)) has the smallest determinant of a matrix with orthogonal rows, and the given top row.
For n = 70 = 2*5*7 = p_1*p_3*p_4, the given basis vector is (1,3,4), and a(70)=78 because the matrix ((1,3,4),(1,1,-1),(-7,5,-2)) has orthogonal rows and determinant 78, which is minimal.
		

Links

• Christopher J. Smyth, Table of n, a(n) for n = 1..6000
• 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 6000

Crossrefs

Cf. A327269 (basis vector is (1,2,...,r)), A327271 (basis vector is (1,1,...,1)), A327272 (basis vector is (1,2,2^2,...,2^{r-1})).

Formula

For n = p_j prime, the matrix is 1 X 1, namely (j), and a(n) = j.

For n = p_{j}*p_{j'}, the matrix is 2 X 2, namely ((j, j'),(-j'/g, j/g)), where g = gcd(j,j'), and a(n) = (j^2 + {j'}^2)/g.

Also easy to see that a(p_{k j_1}*...*p_{k j_r}) = k*a(p_{j_1}*...*p_{j_r}).