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

%I A327267 #33 Dec 15 2024 17:05:57
%S A327267 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,
%T A327267 10,42,11,40,29,50,25,20,12,65,20,24,13,42,14,54,34,82,15,32,8,38,53,
%U A327267 38,16,78,34,114,34,101,17,30,18,122,12,48,15,30
%N A327267 Minimal determinant of a finite-dimensional integer lattice having an orthogonal basis containing a given vector with all entries positive.
%C A327267 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.
%C A327267 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}.
%C A327267 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.
%H A327267 Christopher J. Smyth, <a href="/A327267/b327267.txt">Table of n, a(n) for n = 1..6000</a>
%H A327267 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 A327267 Christopher J. Smyth, <a href="/A327267/a327267.txt">List of n, a(n) and associated matrix for n up to 6000</a>
%F A327267 For n = p_j prime, the matrix is 1 X 1, namely (j), and a(n) = j.
%F A327267 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.
%F A327267 Also easy to see that a(p_{k j_1}*...*p_{k j_r}) = k*a(p_{j_1}*...*p_{j_r}).
%e A327267 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.
%e A327267 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.
%Y A327267 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})).
%K A327267 nonn
%O A327267 1,3
%A A327267 _Christopher J. Smyth_, Aug 31 2019