login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of complete planar branching factorizations of n.
2

%I #5 Sep 12 2018 15:10:42

%S 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,

%T 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,

%U 9,197,2,18,1,9,2,18,1,450,1,2,9,9,2,18,1,225

%N Number of complete planar branching factorizations of n.

%C 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.

%F a(prime^n) = A001003(n - 1).

%F a(product of n distinct primes) = A032037(n).

%e The a(12) = 9 trees:

%e (2*2*3), (2*3*2), (3*2*2),

%e (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).

%t ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]

%t otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];

%t Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

%Y Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.

%K nonn

%O 1,6

%A _Gus Wiseman_, Sep 11 2018