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%I #7 Apr 15 2024 09:47:59
%S 0,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,2,1,2,1,1,2,3,1,1,
%T 1,1,1,1,1,2,1,1,2,1,1,2,1,2,1,2,1,2,1,1,2,1,1,1,1,3,2,1,2,3,1,1,1,1,
%U 1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2,1
%N Maximal length of a set partition of the binary indices of n into blocks all having the same sum.
%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%C If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).
%e The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:
%e {{1,7},{2,6},{3,5}}
%e {{1,5,6},{2,3,7}}
%e {{1,2,3,6},{5,7}}
%e {{1,2,3,5,6,7}}
%e So a(119) = 3.
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Table[Max[Length/@Select[sps[bix[n]],SameQ@@Total/@#&]],{n,0,100}]
%Y Set partitions of this type are counted by A035470, A336137.
%Y A version for factorizations is A371733.
%Y Positions of 1's are A371738.
%Y Positions of terms > 1 are A371784.
%Y A001055 counts factorizations.
%Y A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A070939 gives length of binary expansion.
%Y A321452 counts quanimous partitions, ranks A321454.
%Y A326031 gives weight of the set-system with BII-number n.
%Y A371783 counts k-quanimous partitions.
%Y A371789 counts non-quanimous sets, differences A371790.
%Y A371796 counts quanimous sets, differences A371797.
%Y Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.
%K nonn
%O 0,8
%A _Gus Wiseman_, Apr 14 2024
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