login
A330936
Number of nontrivial factorizations of n into factors > 1.
3
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 2, 0, 3, 0, 5, 0, 0, 0, 7, 0, 0, 0, 5, 0, 3, 0, 2, 2, 0, 0, 10, 0, 2, 0, 2, 0, 5, 0, 5, 0, 0, 0, 9, 0, 0, 2, 9, 0, 3, 0, 2, 0, 3, 0, 14, 0, 0, 2, 2, 0, 3, 0, 10, 3, 0, 0, 9, 0, 0
OFFSET
1,12
COMMENTS
The trivial factorizations of a number are (1) the case with only one factor, and (2) the factorization into prime numbers.
FORMULA
For prime n, a(n) = 0; for nonprime n, a(n) = A001055(n) - 2.
EXAMPLE
The a(n) nontrivial factorizations of n = 8, 12, 16, 24, 36, 48, 60, 72:
(2*4) (2*6) (2*8) (3*8) (4*9) (6*8) (2*30) (8*9)
(3*4) (4*4) (4*6) (6*6) (2*24) (3*20) (2*36)
(2*2*4) (2*12) (2*18) (3*16) (4*15) (3*24)
(2*2*6) (3*12) (4*12) (5*12) (4*18)
(2*3*4) (2*2*9) (2*3*8) (6*10) (6*12)
(2*3*6) (2*4*6) (2*5*6) (2*4*9)
(3*3*4) (3*4*4) (3*4*5) (2*6*6)
(2*2*12) (2*2*15) (3*3*8)
(2*2*2*6) (2*3*10) (3*4*6)
(2*2*3*4) (2*2*18)
(2*3*12)
(2*2*2*9)
(2*2*3*6)
(2*3*3*4)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[DeleteCases[Rest[facs[n]], {_}]], {n, 100}]
CROSSREFS
Positions of nonzero terms are A033942.
Positions of 1's are A030078.
Positions of 2's are A054753.
Nontrivial integer partitions are A007042.
Nontrivial set partitions are A008827.
Nontrivial divisors are A070824.
Sequence in context: A083059 A173440 A348341 * A284687 A046268 A202425
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 04 2020
STATUS
approved