cp's OEIS Frontend

A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

%I A326515 #12 Jan 20 2025 09:07:35
%S A326515 1,1,1,2,1,1,1,3,2,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,2,1,3,1,1,2,1,7,1,1,
%T A326515 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,
%U A326515 1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,2,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2
%N A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.
%C A326515 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 A326515 Antti Karttunen, <a href="/A326515/b326515.txt">Table of n, a(n) for n = 1..65537</a>
%H A326515 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 A326515 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A326515 The a(900) = 9 factorizations:
%e A326515   (3*3*10*10),
%e A326515   (3*3*100), (3*10*30), (9*10*10),
%e A326515   (3*300), (9*100), (10*90), (30*30),
%e A326515   (900).
%t A326515 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326515 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326515 Table[Length[Select[facs[n],SameQ@@Mean/@primeMS/@#&]],{n,100}]
%o A326515 (PARI)
%o A326515 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 A326515 has_same_average_of_pis(facs) = if(!#facs, 1, my(avg=0); for(i=1,#facs,if(!avg, avg=avgpis(facs[i]), if(avg!=avgpis(facs[i]), return(0)))); (1));
%o A326515 A326515(n, m=n, facs=List([])) = if(1==n, has_same_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326515(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A326515 Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326512, A326514, A326516, A326520, A326536.
%K A326515 nonn
%O A326515 1,4
%A A326515 _Gus Wiseman_, Jul 12 2019
%E A326515 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025