A330936 Number of nontrivial factorizations of n into factors > 1.

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.

Links

Table of n, a(n) for n=1..86.

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.

Cf. A001055, A003238, A005121, A317145, A317176, A318812, A330665, A330935.

Sequence in context: A083059 A173440 A348341 * A284687 A046268 A202425

Adjacent sequences: A330933 A330934 A330935 * A330937 A330938 A330939

Keyword

nonn

Author

Gus Wiseman, Jan 04 2020

Status

approved