cp's OEIS Frontend

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

%I A371733 #18 Jan 20 2025 22:52:46
%S A371733 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,2,1,5,1,1,
%T A371733 1,3,1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,6,1,1,1,1,
%U A371733 1,2,1,1,1,1,1,1,1,1,1,1,4,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,4
%N A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.
%C A371733 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. Sum of prime indices is given by A056239.
%C A371733 Factorizations into factors all having the same sum of prime indices are counted by A321455.
%H A371733 Antti Karttunen, <a href="/A371733/b371733.txt">Table of n, a(n) for n = 1..65537</a>
%H A371733 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 A371733 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A371733 The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
%e A371733 The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.
%t A371733 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A371733 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A371733 Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{n,100}]
%o A371733 (PARI)
%o A371733 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A371733 all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
%o A371733 A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A371733 Positions of 1's are A321453, counted by A321451.
%Y A371733 Positions of terms > 1 are A321454, counted by A321452.
%Y A371733 Factorizations of this type are counted by A321455, different sums A321469.
%Y A371733 For different sums instead of same sums we have A371734.
%Y A371733 For set partitions of binary indices we have A371735.
%Y A371733 A001055 counts factorizations.
%Y A371733 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A371733 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371733 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371733 A371789 counts non-quanimous sets, differences A371790.
%Y A371733 A371796 counts quanimous sets, differences A371797.
%Y A371733 Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.
%K A371733 nonn
%O A371733 1,4
%A A371733 _Gus Wiseman_, Apr 13 2024
%E A371733 Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025