A327658 - OEIS

%I #4 Sep 22 2019 08:06:09

%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,4,2,1,3,2,1,1,1,7,1,1,

%T 1,4,1,1,1,4,1,1,1,2,2,1,1,7,2,2,1,2,1,4,1,4,1,1,1,3,1,1,2,11,1,1,1,2,

%U 1,1,1,7,1,1,2,2,1,1,1,7,5,1,1,3,1,1,1

%N Number of factorizations of n that are empty or whose factors have a common divisor > 1.

%C First differs from A319786 at a(900) = 11, A319786(900) = 12.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%e The a(120) = 7 factorizations:

%e   (120)

%e   (2*60)

%e   (4*30)

%e   (6*20)

%e   (10*12)

%e   (2*2*30)

%e   (2*6*10)

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

%t Table[Length[Select[facs[n],#=={}||GCD@@#!=1&]],{n,100}]

%Y See link for additional cross-references.

%Y Cf. A000005, A056239, A112798, A281116, A289509, A327406.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 21 2019