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A319137
Number of strict planar branching factorizations of n.
3
1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 3, 7, 1, 9, 1, 9, 3, 3, 1, 37, 1, 3, 3, 9, 1, 25, 1, 21, 3, 3, 3, 57, 1, 3, 3, 37, 1, 25, 1, 9, 9, 3, 1, 161, 1, 9, 3, 9, 1, 37, 3, 37, 3, 3, 1, 153, 1, 3, 9, 75, 3, 25, 1, 9, 3, 25, 1, 345, 1, 3, 9, 9, 3, 25, 1, 161
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.
FORMULA
a(prime^n) = A319123(n + 1).
a(product of n distinct primes) = A319122(n).
EXAMPLE
The a(12) = 9 trees:
12,
(2*6), (3*4), (4*3),(6*2),
(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[sotfs[n]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 11 2018
STATUS
approved