A301829 - OEIS

%I #5 Apr 16 2018 18:52:19

%S 0,1,1,3,1,4,1,7,3,4,1,12,1,4,4,15,1,12,1,12,4,4,1,29,3,4,7,12,1,17,1,

%T 29,4,4,4,37,1,4,4,29,1,17,1,12,12,4,1,64,3,12,4,12,1,29,4,29,4,4,1,

%U 53,1,4,12,54,4,17,1,12,4,17,1,92,1,4,12,12,4,17

%N Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

%F a(n) = Sum_{d|n, d>1} f(d) * f(n/d) where f(n) = A001055(n) is the number of factorizations of n into factors greater than 1.

%e The a(12) = 12 submultisets ("<" means subset or equal):

%e (2)<(2*2*3), (3)<(2*2*3), (2*2)<(2*2*3), (2*3)<(2*2*3), (2*2*3)<(2*2*3),

%e (2)<(2*6), (6)<(2*6), (2*6)<(2*6),

%e (3)<(3*4), (4)<(3*4), (3*4)<(3*4),

%e (12)<(12).

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

%t Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Rest[Divisors[n]]}],{n,100}]

%Y Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301830.

%K nonn

%O 1,4

%A _Gus Wiseman_, Mar 27 2018