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A327267
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Minimal determinant of a finite-dimensional integer lattice having an orthogonal basis containing a given vector with all entries positive.
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6
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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}).
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EXAMPLE
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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.
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CROSSREFS
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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)).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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