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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.

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.

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(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.

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.

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

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.

Any integer k in the sequence encodes (by 'Heinz encoding' cf. A056239) a multiset of integers whose gcd is 1, namely the multiset containing r_j copies of j if k factors as Product_j prime(j)^{r_j} with gcd_j j = 1.

Clearly the sequence contains all even numbers and no odd primes or odd prime powers. It also clearly contains all numbers that are divisible by consecutive primes.

The sequence is the list of those k such that A289508(k) = 1.

It is also the list of those k such that A289506(k) = A289507(k).

Heinz numbers of integer partitions with relatively prime parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 13 2018

The number n = Product_j p_j can be regarded as an index for the multiset of all the j's, occurring with multiplicity corresponding to the highest power of p_j dividing n. Then a(n) is the gcd of the elements of this multiset. Compare A056239, where the same encoding for integer multisets('Heinz encoding') is used, but where A056239(n) is the sum, rather than the gcd, of the elements of the corresponding multiset (partition) of the j's. Cf. also A003963, for which A003963(n) is the product of the elements of the corresponding multiset.

a(m*n) = gcd(a(m),a(n)). - Robert Israel, Jul 19 2017

Given an integer linear equation Sum_{j=1..k} e_j x_j = 0, a(n) is also the modulus of the determinant whose first row is e_1, e_2, ..., e_k and whose other k-1 rows form an integral basis for the integer solution space of the equation. Here n = Product_j p_{e_j}, where p_{e_j} is the e_j-th prime.

For the proof, see Links.

Also a(n) = A289506(n) when gcd_j e_j = 1, which occurs for the numbers n in A289509.