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

1,3

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.

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

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}).

Example

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.

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)).

Sequence in context: A302081 A209147 A355059 * A328666 A036716 A026399

Adjacent sequences: A327264 A327265 A327266 * A327268 A327269 A327270

Keyword

nonn

Author

Christopher J. Smyth, Aug 31 2019

Status

approved