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A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.

%I A371784 #5 Apr 16 2024 19:21:42
%S A371784 7,13,15,22,25,27,30,31,39,42,45,47,49,51,54,59,60,62,63,75,76,82,85,
%T A371784 87,90,93,94,95,97,99,102,107,108,109,110,115,117,119,120,122,125,126,
%U A371784 127,141,143,147,148,153,155,158,162,165,167,170,173,175,179,180
%N A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.
%C A371784 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.
%e A371784 The binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is in the sequence.
%e A371784 The terms together with their binary expansions and binary indices begin:
%e A371784    7:     111 ~ {1,2,3}
%e A371784   13:    1101 ~ {1,3,4}
%e A371784   15:    1111 ~ {1,2,3,4}
%e A371784   22:   10110 ~ {2,3,5}
%e A371784   25:   11001 ~ {1,4,5}
%e A371784   27:   11011 ~ {1,2,4,5}
%e A371784   30:   11110 ~ {2,3,4,5}
%e A371784   31:   11111 ~ {1,2,3,4,5}
%e A371784   39:  100111 ~ {1,2,3,6}
%e A371784   42:  101010 ~ {2,4,6}
%e A371784   45:  101101 ~ {1,3,4,6}
%e A371784   47:  101111 ~ {1,2,3,4,6}
%e A371784   49:  110001 ~ {1,5,6}
%e A371784   51:  110011 ~ {1,2,5,6}
%e A371784   54:  110110 ~ {2,3,5,6}
%e A371784   59:  111011 ~ {1,2,4,5,6}
%e A371784   60:  111100 ~ {3,4,5,6}
%e A371784   62:  111110 ~ {2,3,4,5,6}
%e A371784   63:  111111 ~ {1,2,3,4,5,6}
%t A371784 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371784 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371784 Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]>1&]
%Y A371784 Set partitions with all equal block-sums are counted by A035470.
%Y A371784 Positions of terms > 1 in A336137 and A371735.
%Y A371784 The complement is A371738.
%Y A371784 A000110 counts set partitions.
%Y A371784 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A371784 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A371784 A070939 gives length of binary expansion.
%Y A371784 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371784 A321452 counts quanimous partitions, ranks A321454.
%Y A371784 A371789 counts non-quanimous sets, differences A371790.
%Y A371784 A371796 counts quanimous sets, differences A371797.
%Y A371784 Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.
%K A371784 nonn,base
%O A371784 1,1
%A A371784 _Gus Wiseman_, Apr 16 2024