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Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.
2

%I #5 Apr 16 2024 19:21:42

%S 7,13,15,22,25,27,30,31,39,42,45,47,49,51,54,59,60,62,63,75,76,82,85,

%T 87,90,93,94,95,97,99,102,107,108,109,110,115,117,119,120,122,125,126,

%U 127,141,143,147,148,153,155,158,162,165,167,170,173,175,179,180

%N Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with 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.

%e 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 The terms together with their binary expansions and binary indices begin:

%e 7: 111 ~ {1,2,3}

%e 13: 1101 ~ {1,3,4}

%e 15: 1111 ~ {1,2,3,4}

%e 22: 10110 ~ {2,3,5}

%e 25: 11001 ~ {1,4,5}

%e 27: 11011 ~ {1,2,4,5}

%e 30: 11110 ~ {2,3,4,5}

%e 31: 11111 ~ {1,2,3,4,5}

%e 39: 100111 ~ {1,2,3,6}

%e 42: 101010 ~ {2,4,6}

%e 45: 101101 ~ {1,3,4,6}

%e 47: 101111 ~ {1,2,3,4,6}

%e 49: 110001 ~ {1,5,6}

%e 51: 110011 ~ {1,2,5,6}

%e 54: 110110 ~ {2,3,5,6}

%e 59: 111011 ~ {1,2,4,5,6}

%e 60: 111100 ~ {3,4,5,6}

%e 62: 111110 ~ {2,3,4,5,6}

%e 63: 111111 ~ {1,2,3,4,5,6}

%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]>1&]

%Y Set partitions with all equal block-sums are counted by A035470.

%Y Positions of terms > 1 in A336137 and A371735.

%Y The complement is A371738.

%Y A000110 counts set partitions.

%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 A321451 counts non-quanimous partitions, ranks A321453.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y A371789 counts non-quanimous sets, differences A371790.

%Y A371796 counts quanimous sets, differences A371797.

%Y Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

%K nonn,base

%O 1,1

%A _Gus Wiseman_, Apr 16 2024