OFFSET
1,4
COMMENTS
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
FORMULA
EXAMPLE
The a(216) = 28 factorizations:
(2*2*2*3*3*3) (2*2*2*3*9) (2*2*6*9) (3*8*9) (3*72) (216)
(2*2*3*3*6) (2*3*4*9) (4*6*9) (4*54)
(2*3*3*3*4) (2*3*6*6) (2*2*54) (6*36)
(3*3*4*6) (2*3*36) (8*27)
(2*2*3*18) (2*4*27) (9*24)
(2*3*3*12) (2*6*18) (12*18)
(2*9*12) (2*108)
(3*3*24)
(3*4*18)
(3*6*12)
The a(270) = 20 factorizations:
(2*3*3*3*5) (2*3*5*9) (5*6*9) (3*90) (270)
(3*3*5*6) (2*3*45) (5*54)
(2*3*3*15) (2*5*27) (6*45)
(2*9*15) (9*30)
(3*3*30) (10*27)
(3*5*18) (15*18)
(3*6*15) (2*135)
(3*9*10)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m], !MatchQ[#, {___, x_, x_, ___}]&]!={};
Table[Length[Select[facs[n], MatchQ[#, {x_, x_}]||sepQ[#]&]], {n, 100}]
CROSSREFS
Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Positions of 2's are A323644.
The complement is counted by A348381.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 30 2021
STATUS
approved