cp's OEIS Frontend

A321455 Number of ways to factor n into factors > 1 all having the same sum of prime indices.

%I A321455 #19 Jan 20 2025 13:09:23
%S A321455 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,1,1,1,1,1,1,1,2,1,2,1,1,2,1,2,1,1,
%T A321455 1,3,1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,1,1,1,1,
%U A321455 1,2,1,1,1,1,1,1,1,1,1,1,3,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3
%N A321455 Number of ways to factor n into factors > 1 all having the same sum of prime indices.
%C A321455 Also the number of multiset partitions of the multiset of prime indices of n with equal block-sums.
%C A321455 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. The sum of prime indices of n is A056239(n).
%H A321455 Antti Karttunen, <a href="/A321455/b321455.txt">Table of n, a(n) for n = 1..65537</a>
%H A321455 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 A321455 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A321455 The a(1440) = 6 factorizations into factors all having the same sum of prime indices:
%e A321455   (10*12*12)
%e A321455   (5*6*6*8)
%e A321455   (9*10*16)
%e A321455   (30*48)
%e A321455   (36*40)
%e A321455   (1440)
%e A321455 The a(900) = 5 multiset partitions with equal block-sums:
%e A321455   {{1,1,2,2,3,3}}
%e A321455   {{3,3},{1,1,2,2}}
%e A321455   {{1,2,3},{1,2,3}}
%e A321455   {{1,3},{1,3},{2,2}}
%e A321455   {{3},{3},{1,2},{1,2}}
%t A321455 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A321455 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A321455 Table[Length[Select[facs[n],SameQ@@hwt/@#&]],{n,100}]
%o A321455 (PARI)
%o A321455 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A321455 all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
%o A321455 A321455(n, m=n, facs=List([])) = if(1==n, all_have_same_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321455(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A321455 Positions of 1's are A321453. Positions of terms > 1 are A321454.
%Y A321455 Cf. A001055, A035470, A056239, A279787, A305551, A321469, A322794, A326515, A326516, A326518, A326534.
%K A321455 nonn
%O A321455 1,4
%A A321455 _Gus Wiseman_, Nov 10 2018
%E A321455 Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025