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

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

Author

Christopher J. Smyth, Oct 24 2019

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A327267.