cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A327273 An upper bound sequence for A327272.

Original entry on oeis.org

1, 5, 105, 425, 144925, 580125, 243697125, 46420625, 253459513125
Offset: 0

Views

Author

Christopher J. Smyth, Sep 09 2019

Keywords

Comments

a(n) can be calculated by a faster algorithm than that for A327272(n). It gives a small (but not necessarily smallest) positive determinant with top row [1,2,2^2,...,2^n] and all entries integers, and rows orthogonal. Note that a(n) = A327272(n) for n=0,1 and 3. See Pinner and Smyth link below for both algorithms, and more details of the sequences.

Examples

			a(2) = 105 since the algorithm for a(n) produces the determinant([[1,2,4],[2,-1,0],[4,8,-5]]) = 105, having top row [1,2,2^2] and all rows orthogonal.

Links

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

Original entry on oeis.org

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

Views

Author

Christopher J. Smyth, Aug 31 2019

Keywords

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

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}).

A327269 Smallest modulus of any integer n X n determinant with top row 1,2,...,n and rows nonzero and pairwise orthogonal.

Original entry on oeis.org

1, 5, 42, 90, 990, 5733, 6720, 39168
Offset: 1

Views

Author

Christopher J. Smyth, Sep 02 2019

Keywords

Comments

Also a(n) = A327267(p_1...p_n), with p_j = j-th prime, since p_1...p_n is the Heinz code for the multiset {1,2,3,...,n}. For more details see Pinner and Smyth link.

Examples

			a(3)=42 since det([1,2,3],[1,-2,1],[4,1,-2]) = 42 and is the smallest positive determinant with top row [1,2,3] and all rows orthogonal.

Links

Crossrefs

Subsequence of A327267-- see comments; A327271 is similar, but determinant's top row is n 1's; A327272 is similar, but determinant's top row is 1,2,2^2,...,2^n.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.