cp's OEIS Frontend

A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

%I A326516 #11 Jan 20 2025 09:10:36
%S A326516 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 A326516 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 A326516 2,5,1,8,1,2,3,3,2,5,1,5,1,2,1,8,2,2,2,4,1,7,2,3,2,2,2,6,1,3,3,5,1,5,1,4,4
%N A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.
%C A326516 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 A326516 Antti Karttunen, <a href="/A326516/b326516.txt">Table of n, a(n) for n = 1..65537</a>
%H A326516 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%H A326516 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A326516 The a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).
%t A326516 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326516 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326516 Table[Length[Select[facs[n],UnsameQ@@Mean/@primeMS/@#&]],{n,100}]
%o A326516 (PARI)
%o A326516 avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
%o A326516 all_have_different_average_of_pis(facs) = if(!#facs, 1, (#Set(apply(avgpis,facs)) == #facs));
%o A326516 A326516(n, m=n, facs=List([])) = if(1==n, all_have_different_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326516(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A326516 Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.
%K A326516 nonn
%O A326516 1,6
%A A326516 _Gus Wiseman_, Jul 12 2019
%E A326516 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025