cp's OEIS Frontend

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

%I A371734 #12 Jan 20 2025 13:09:31
%S A371734 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,3,1,2,2,2,
%T A371734 2,3,1,2,2,3,1,3,1,2,2,2,1,3,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,3,2,3,1,2,
%U A371734 2,3,1,3,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2,3,1,3,2,2,2,2,2,3,1,2,2,3,1,3,1,3,3
%N A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.
%C A371734 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 A371734 Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.
%H A371734 Antti Karttunen, <a href="/A371734/b371734.txt">Table of n, a(n) for n = 1..65537</a>
%H A371734 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 A371734 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A371734 The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.
%t A371734 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A371734 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A371734 Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]
%o A371734 (PARI)
%o A371734 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A371734 all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
%o A371734 A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_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,A371734(n/d, d, newfacs)))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A371734 For set partitions of binary indices we have A000120, same sums A371735.
%Y A371734 Positions of 1's are A000430.
%Y A371734 Positions of terms > 1 are A080257.
%Y A371734 Factorizations of this type are counted by A321469, same sums A321455.
%Y A371734 For same instead of different sums we have A371733.
%Y A371734 A001055 counts factorizations.
%Y A371734 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A371734 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371734 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371734 A321452 counts quanimous partitions, ranks A321454.
%Y A371734 Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.
%K A371734 nonn
%O A371734 1,6
%A A371734 _Gus Wiseman_, Apr 13 2024
%E A371734 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025