A326516 - OEIS

%I #5 Jul 12 2019 20:20:04

%S 1,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,3,1,3,2,2,1,4,1,2,1,3,1,4,1,1,2,2,

%T 2,5,1,2,2,4,1,5,1,3,3,2,1,5,1,3,2,3,1,4,2,4,2,2,1,8,1,2,3,1,2,5,1,3,

%U 2,5,1,8,1,2,3,3,2,5,1,5,1,2,1,8,2,2,2

%N Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

%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.

%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>

%e The a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%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],UnsameQ@@Mean/@primeMS/@#&]],{n,100}]

%Y Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jul 12 2019