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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 04 2020
STATUS
approved