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

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, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 48, 50, 52, 53, 55, 56, 57, 58, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..67.

Example

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.
The terms together with their binary expansions and binary indices begin:
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  10:  1010 ~ {2,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  23: 10111 ~ {1,2,3,5}

Mathematica

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Select[Range[100], Length[Select[sps[bix[#]], SameQ@@Total/@#&]]==1&]

Crossrefs

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

Positions of 1's in A336137 and A371735.

The complement is A371784.

A000110 counts set partitions.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A070939 gives length of binary expansion.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

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

Sequence in context: A333635 A364379 A102799 * A059519 A163101 A157465

Adjacent sequences: A371735 A371736 A371737 * A371739 A371740 A371741

Keyword

nonn,base

Author

Gus Wiseman, Apr 14 2024

Status

approved