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A recursively defined integer-valued function of integer multisets.
1

%I #7 Nov 16 2019 14:11:34

%S 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,

%T 10,70,11,40,29,50,25,20,12,65,20,24,13,105,14,54,34,82,15,32,8,38,53,

%U 38,16,78,34,114,34,101,17,30,18,122,12,48,15

%N A recursively defined integer-valued function of integer multisets.

%C For singletons {k}, F({k}) = k. For multisets {k_1,...,k_r} with r>1, F is defined recursively by

%C 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

%C {k_1,...,k_r} = {k'_1,...,k'_{r'}} union {k"_1,...,k"_{r-r'}}, where 1 <= r' < r.

%C 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.

%C 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).

%C Also a(1)=0.

%C 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}).

%C See the Pinner/Smyth link for the construction of these matrices.

%H Chris Pinner and Chris Smyth, <a href="https://www.maths.ed.ac.uk/~chris/papers/MinimalLattices040919.pdf">Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector</a>

%e 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.

%Y Cf. A327267.

%K nonn

%O 1,3

%A _Christopher J. Smyth_, Oct 24 2019