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A319136
Number of complete planar branching factorizations of n.
2
0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 2, 11, 1, 9, 1, 9, 2, 2, 1, 44, 1, 2, 3, 9, 1, 18, 1, 45, 2, 2, 2, 66, 1, 2, 2, 44, 1, 18, 1, 9, 9, 2, 1, 225, 1, 9, 2, 9, 1, 44, 2, 44, 2, 2, 1, 132, 1, 2, 9, 197, 2, 18, 1, 9, 2, 18, 1, 450, 1, 2, 9, 9, 2, 18, 1, 225
OFFSET
1,6
COMMENTS
A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.
FORMULA
a(prime^n) = A001003(n - 1).
a(product of n distinct primes) = A032037(n).
EXAMPLE
The a(12) = 9 trees:
(2*2*3), (2*3*2), (3*2*2),
(2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
MATHEMATICA
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f], {f, Select[ordfacs[n], Length[#]>1&]}], n];
Table[Length[Select[otfs[n], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 11 2018
STATUS
approved