%I #4 Jan 05 2020 08:11:24

%S 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,

%T 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,

%U 0,3,0,14,0,0,2,2,0,3,0,10,3,0,0,9,0,0

%N Number of nontrivial factorizations of n into factors > 1.

%C The trivial factorizations of a number are (1) the case with only one factor, and (2) the factorization into prime numbers.

%F For prime n, a(n) = 0; for nonprime n, a(n) = A001055(n) - 2.

%e The a(n) nontrivial factorizations of n = 8, 12, 16, 24, 36, 48, 60, 72:

%e (2*4) (2*6) (2*8) (3*8) (4*9) (6*8) (2*30) (8*9)

%e (3*4) (4*4) (4*6) (6*6) (2*24) (3*20) (2*36)

%e (2*2*4) (2*12) (2*18) (3*16) (4*15) (3*24)

%e (2*2*6) (3*12) (4*12) (5*12) (4*18)

%e (2*3*4) (2*2*9) (2*3*8) (6*10) (6*12)

%e (2*3*6) (2*4*6) (2*5*6) (2*4*9)

%e (3*3*4) (3*4*4) (3*4*5) (2*6*6)

%e (2*2*12) (2*2*15) (3*3*8)

%e (2*2*2*6) (2*3*10) (3*4*6)

%e (2*2*3*4) (2*2*18)

%e (2*3*12)

%e (2*2*2*9)

%e (2*2*3*6)

%e (2*3*3*4)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[DeleteCases[Rest[facs[n]],{_}]],{n,100}]

%Y Positions of nonzero terms are A033942.

%Y Positions of 1's are A030078.

%Y Positions of 2's are A054753.

%Y Nontrivial integer partitions are A007042.

%Y Nontrivial set partitions are A008827.

%Y Nontrivial divisors are A070824.

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

%K nonn

%O 1,12

%A _Gus Wiseman_, Jan 04 2020