# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a371734 Showing 1-1 of 1 %I A371734 #7 Apr 14 2024 03:49:55 %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 %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 Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict. %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}] %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE