A371734 - OEIS

%I #7 Apr 14 2024 03:49:55

%S 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 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 2,3,1,3,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2

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

%C 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 Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>

%e 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 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];

%t Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

%Y For set partitions of binary indices we have A000120, same sums A371735.

%Y Positions of 1's are A000430.

%Y Positions of terms > 1 are A080257.

%Y Factorizations of this type are counted by A321469, same sums A321455.

%Y For same instead of different sums we have A371733.

%Y A001055 counts factorizations.

%Y A002219 (aerated) counts biquanimous partitions, ranks A357976.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

%Y A321451 counts non-quanimous partitions, ranks A321453.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

%K nonn

%O 1,6

%A _Gus Wiseman_, Apr 13 2024