cp's OEIS Frontend

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

%I A371738 #8 Apr 16 2024 19:21:46
%S A371738 1,2,3,4,5,6,8,9,10,11,12,14,16,17,18,19,20,21,23,24,26,28,29,32,33,
%T A371738 34,35,36,37,38,40,41,43,44,46,48,50,52,53,55,56,57,58,61,64,65,66,67,
%U A371738 68,69,70,71,72,73,74,77,78,79,80,81,83,84,86,88,89,91,92
%N A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.
%C A371738 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 A371738 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 not in the sequence.
%e A371738 The terms together with their binary expansions and binary indices begin:
%e A371738    1:     1 ~ {1}
%e A371738    2:    10 ~ {2}
%e A371738    3:    11 ~ {1,2}
%e A371738    4:   100 ~ {3}
%e A371738    5:   101 ~ {1,3}
%e A371738    6:   110 ~ {2,3}
%e A371738    8:  1000 ~ {4}
%e A371738    9:  1001 ~ {1,4}
%e A371738   10:  1010 ~ {2,4}
%e A371738   11:  1011 ~ {1,2,4}
%e A371738   12:  1100 ~ {3,4}
%e A371738   14:  1110 ~ {2,3,4}
%e A371738   16: 10000 ~ {5}
%e A371738   17: 10001 ~ {1,5}
%e A371738   18: 10010 ~ {2,5}
%e A371738   19: 10011 ~ {1,2,5}
%e A371738   20: 10100 ~ {3,5}
%e A371738   21: 10101 ~ {1,3,5}
%e A371738   23: 10111 ~ {1,2,3,5}
%t A371738 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371738 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371738 Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]==1&]
%Y A371738 Set partitions with all equal block-sums are counted by A035470.
%Y A371738 Positions of 1's in A336137 and A371735.
%Y A371738 The complement is A371784.
%Y A371738 A000110 counts set partitions.
%Y A371738 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A371738 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A371738 A070939 gives length of binary expansion.
%Y A371738 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371738 A321452 counts quanimous partitions, ranks A321454.
%Y A371738 A371789 counts non-quanimous sets, differences A371790.
%Y A371738 A371796 counts quanimous sets, differences A371797.
%Y A371738 Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.
%K A371738 nonn,base
%O A371738 1,2
%A A371738 _Gus Wiseman_, Apr 14 2024