A327267 -id:A327267 - cp's OEIS frontend

A348557 List of indices k such that A327267(k) is less than A328666(k).

30, 42, 70, 90, 102, 120, 154, 165, 170, 180, 182, 198, 210, 220, 224, 230, 238, 255, 264, 270, 273, 285, 286, 306, 308, 322, 330, 340, 350, 360, 374, 380, 385, 399, 406, 418, 420, 434, 435, 442, 465, 480, 486, 490, 494, 495, 506, 510, 518, 525, 544, 546, 550, 552, 558

Offset: 1

Christopher J. Smyth, Oct 22 2021

nonn

It is known that A327267(n) <= A328666(n) for all n. See link. For n up to 6000, the inequality is strict only for 1193 values of n.

Indeed, the purpose of A328666 is to give a good upper bound sequence for the sequence A327267, while being much easier to compute.

			a(1) = 30 since 30 is the first value of k for which A327267(k) < A328666(k).

Christopher J. Smyth, Table of n, a(n) for n = 1..1193
Chris Pinner and Chris Smyth, Lattices of minimal index in Z^n having an orthogonal basis containing a given vector, preprint, 2021.

Cf. A327267 and A328666.

A327271 Smallest modulus of any n X n integer determinant whose top row has all 1's and whose rows are pairwise orthogonal.

Original entry on oeis.org

1, 2, 6, 8, 40, 48, 336, 128, 864, 1280, 8448, 3072, 39936

Offset: 1

Christopher J. Smyth, Sep 02 2019

a(n) = A327267(2^n), since 2^n = (p_1)^n is the Heinz code for the multiset {1,1,...,1}.
See Pinner and Smyth link below for more details, including an algorithm for computing A327267(n). Also, see file link below for {(n, a(n), matrix(n)), n <= 13}, where matrix(n) has minimal modulus determinant equal to a(n) among n X n matrices with top row all 1's and all rows orthogonal.
For the first 13 terms, the number of prime factors counted with multiplicity equals n-1: A001222(a(n))=n-1. How far does this hold? - Jon Maiga, Sep 07 2019

			a(3) = 6 because the matrix [[1,1,1],[1,-1,0],[1,1,-2]] has top row of 3 1's and all rows orthogonal, and minimal positive determinant equal to 6.

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 13

Subsequence of A327267, see comments; A327273 is similar, but determinant's top row is 1,2,2^2,...,2^{n-1}.

A327272 Smallest modulus of any (n+1) X (n+1) integer determinant whose top row is 1,2,2^2,...,2^n and whose rows are pairwise orthogonal.

Original entry on oeis.org

1, 5, 42, 425, 17050, 54600, 11468100

Offset: 1

Christopher J. Smyth, Sep 09 2019

An algorithm for generating a(n) is given in the Pinner and Smyth link, where more details about a(n) can be found.

Also, see file link below for {(n,a(n),matrix(n)),0 <= n <= 6}, where matrix(n) has minimal modulus determinant equal to a(n) among (n+1) X (n+1) matrices with top row 1,2,2^2,...,2^n and all rows orthogonal.

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

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 0 <= n <= 6

Subsequence of A327267-- see comments; A327269 is similar, but determinant's top row is 1,2,...,n; A327271 is similar, but determinant's top row consists of n 1's.

a(n) = A327267(Product_{k=0..n} prime(2^k)) = A327267(A325782(n+1)).

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

Christopher J. Smyth, Sep 02 2019

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.

			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.

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 <= 8

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.

A328666 A recursively defined integer-valued function of integer multisets.

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, 70, 11, 40, 29, 50, 25, 20, 12, 65, 20, 24, 13, 105, 14, 54, 34, 82, 15, 32, 8, 38, 53, 38, 16, 78, 34, 114, 34, 101, 17, 30, 18, 122, 12, 48, 15

Offset: 1

Christopher J. Smyth, Oct 24 2019

nonn

For singletons {k}, F({k}) = k. For multisets {k_1,...,k_r} with r>1, F is defined recursively by
F({k_1,...,k_r}) = min F({k'1,...,k'{r'}})*F({k"1,...,k"{r-r'}})*K/g, where the minimum is taken over all 2-partitions
{k_1,...,k_r} = {k'1,...,k'{r'}} union {k"1,...,k"{r-r'}}, where 1 <= r' < r.
Here K = Sum_{i=1..r} {k_i}^2 and g = gcd(K"*k'1,...,K"*k'{r'},K'*k"1,...,K'*k"{r-r'}), where K' = Sum_{i=1..r'} {k'i}^2 and K" = Sum{i=1..(r-r')} {k"_i}^2.
The function F is then encoded as an integer sequence by a(n)= F({k_1,..,k_r}), where n=p_{k_1}p_{k_2}..p_{k_r},  p_k being the k-th prime (Heinz encoding).
Also a(1)=0.
The significance of this sequence is that for given multiset {k_1,...,k_r} there is an r X r integer matrix with all rows pairwise orthogonal whose top row is {k_1,...,k_r} and whose determinant is F({k_1,...,k_r}).
See the Pinner/Smyth link for the construction of these matrices.

			For r=2 only allowable 2-partition of {k_1,k_2} is {k_1} union {k_2}, giving K = {k_1}^2+{k_2}^2, K' = {k_1}^2, K" = {k_2}^2, g = k_1*k_2*gcd(k_1,k_2), n =  p_{k_1}p_{k_2}, F({k_i}) = k_i (i=1,2), and so a(n) = F({k_1,k_2}) = F({k_1})F({k_2})K/g = ({k_1}^2+{k_2}^2)/gcd(k_1,k_2). Thus for example a(10) = a(p_1p_3) = 1^2+3^2 = 10.

Chris Pinner and Chris Smyth, Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector

Cf. A327267.

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A348557 List of indices k such that A327267(k) is less than A328666(k).

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A327271 Smallest modulus of any n X n integer determinant whose top row has all 1's and whose rows are pairwise orthogonal.

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A327272 Smallest modulus of any (n+1) X (n+1) integer determinant whose top row is 1,2,2^2,...,2^n and whose rows are pairwise orthogonal.

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

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A328666 A recursively defined integer-valued function of integer multisets.

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