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%I #29 Oct 18 2019 03:40:21
%S 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 10,42,11,40,29,50,25,20,12,65,20,24,13,42,14,54,34,82,15,32,8,38,53,
%U 38,16,78,34,114,34,101,17,30,18,122,12,48,15,30
%N Minimal determinant of a finite-dimensional integer lattice having an orthogonal basis containing a given vector with all entries positive.
%C 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 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 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 Christopher J. Smyth, <a href="/A327267/b327267.txt">Table of n, a(n) for n = 1..6000</a>
%H 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 Christopher J. Smyth, <a href="/A327267/a327267.txt">List of n, a(n) and associated matrix for n up to 6000</a>
%F For n = p_j prime, the matrix is 1 X 1, namely (j), and a(n) = j.
%F 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 Also easy to see that a(p_{k j_1}*...*p_{k j_r}) = k*a(p_{j_1}*...*p_{j_r}).
%e 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 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 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 nonn
%O 1,3
%A _Christopher J. Smyth_, Aug 31 2019
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