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A325988
Number of covering (or complete) factorizations of n.
7
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
OFFSET
1,8
COMMENTS
First differs from A072911 at a(64) = 5, A072911(64) = 4.
A covering factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of some submultiset of the factors.
FORMULA
a(2^n) = A126796(n).
EXAMPLE
The a(64) = 5 factorizations:
(2*2*2*2*2*2)
(2*2*2*2*4)
(2*2*2*8)
(2*2*4*4)
(2*4*8)
The a(96) = 4 factorizations:
(2*2*2*2*2*3)
(2*2*2*3*4)
(2*2*3*8)
(2*3*4*4)
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], Union[Times@@@Subsets[#]]==Divisors[n]&]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2019
STATUS
approved