OFFSET
1,3
COMMENTS
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.
LINKS
Chris Pinner and Chris Smyth, Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Christopher J. Smyth, Oct 24 2019
STATUS
approved