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A325989
Number of perfect factorizations of n.
3
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,8
COMMENTS
A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).
FORMULA
a(2^n) = A002033(n).
EXAMPLE
The a(216) = 4 perfect factorizations:
(2*2*2*3*3*3)
(2*2*2*3*9)
(2*3*3*3*4)
(2*3*4*9)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]], {n, 100}]
CROSSREFS
Positions of terms > 1 are A325990.
Sequence in context: A359411 A367516 A368979 * A055229 A367698 A367931
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2019
STATUS
approved