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A319138
Number of complete strict planar branching factorizations of n.
3
0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 4, 1, 2, 2, 0, 1, 4, 1, 4, 2, 2, 1, 8, 0, 2, 0, 4, 1, 18, 1, 0, 2, 2, 2, 28, 1, 2, 2, 8, 1, 18, 1, 4, 4, 2, 1, 16, 0, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 84, 1, 2, 4, 0, 2, 18, 1, 4, 2, 18, 1, 112, 1, 2, 4, 4, 2, 18, 1, 16, 0, 2, 1
OFFSET
1,6
COMMENTS
A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n. A strict planar branching factorization is complete if the leaves are all prime numbers.
FORMULA
a(prime^n) = A000007(n - 1).
a(product of n distinct primes) = A032037(n).
EXAMPLE
The a(12) = 4 trees: (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).
MATHEMATICA
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f], {f, Select[ordfacs[n], And[Length[#]>1, UnsameQ@@#]&]}], n];
Table[Length[Select[sotfs[n], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 11 2018
STATUS
approved