A318577 Number of complete multimin tree-factorizations of n.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 11, 1, 3, 1, 4, 1, 1, 1, 19, 1, 1, 3, 4, 1, 4, 1, 45, 1, 1, 1, 17, 1, 1, 1, 19, 1, 4, 1, 4, 4, 1, 1, 96, 1, 3, 1, 4, 1, 11, 1, 19, 1, 1, 1, 26, 1, 1, 4, 197, 1, 4, 1, 4, 1, 4, 1, 104, 1, 1, 3, 4, 1, 4, 1, 96, 11, 1, 1, 26, 1, 1, 1, 19, 1, 19, 1, 4, 1, 1, 1, 501, 1, 3, 4, 17

Offset

1,8

Comments

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

Links

Table of n, a(n) for n=1..100.

Formula

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

a(product of n distinct primes) = A000311(n).

Example

The a(12) = 4 trees are (2*2*3), (2*(2*3)), ((2*3)*2), ((2*2)*3).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n], Length[#]>1&], OrderedQ[FactorInteger[#][[1, 1]]&/@#]&]), n];
Table[Length[Select[mmftrees[n], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {n, 100}]

Crossrefs

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A319118, A319121.

Sequence in context: A157261 A079110 A079619 * A157603 A305439 A270825

Adjacent sequences: A318574 A318575 A318576 * A318578 A318579 A318580

Keyword

nonn

Author

Gus Wiseman, Sep 11 2018

Status

approved