OFFSET
1,4
COMMENTS
An integer factorization of n is a multiset of positive integers > 1 with product n.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Conjecture: 9 is missing from this sequence.
EXAMPLE
The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
(2) (4) (16) (24) (48) (72)
(2*2) (4*4) (2*2*6) (3*4*4) (2*6*6)
(2*2*4) (2*2*2*3) (2*2*12) (3*3*8)
(2*2*2*2) (2*2*2*6) (2*2*18)
(2*2*3*4) (2*2*2*9)
(2*2*2*2*3) (2*2*3*6)
(2*3*3*4)
(2*2*2*3*3)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n], Length[modes[#]]==1&]], {n, 100}]
CROSSREFS
A089723 counts constant factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 27 2023
STATUS
approved