A326976 - OEIS

%I #7 Aug 13 2019 13:20:08

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,5,1,1,

%T 1,2,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,7,1,1,1,1,

%U 1,1,1,5,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1

%N Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.

%e The a(72) = 5 factorizations:

%e   (3*4*6)

%e   (2*3*12)

%e   (2*2*3*6)

%e   (2*3*3*4)

%e   (2*2*2*3*3)

%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],n==1||Union[Select[GCD@@@Rest[Subsets[#]],PrimeQ]]==First/@FactorInteger[n]&]],

%t {n,100}]

%Y Factorizations whose dual is a weak antichain are A326975.

%Y T_1 factorizations (whose dual is a strict antichain) are A327012.

%Y T_0 factorizations (whose dual is strict) are A316978.

%Y Cf. A001055, A326947, A326965, A326972, A326974, A326977, A326979.

%K nonn

%O 1,8

%A _Gus Wiseman_, Aug 13 2019