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A328666 A recursively defined integer-valued function of integer multisets. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A327267.
Sequence in context: A209147 A355059 A327267 * A036716 A026399 A117267
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 1 06:29 EST 2024. Contains 370430 sequences. (Running on oeis4.)