A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

7, 1, 4, 20, 68, 320, 352, 1088, 3136, 13376, 16704, 5184, 82240, 70720, 17472

Offset

0,1

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Links

Table of n, a(n) for n=0..14.

Wikipedia, Axiom of choice.

Example

The terms together with the corresponding set-systems begin:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{1,2}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
    352: {{2,3},{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
   3136: {{1,2,3},{1,2,4},{3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
c=Table[Length[Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]], {n, 1000}];
spnm[y_]:=Max@@NestWhile[Most, y, Union[#]!=Range[0, Max@@#]&];
Table[Position[c, n][[1, 1]], {n, 0, spnm[c]}]

Crossrefs

Positions of first appearances in A367905.

The sorted version is A367911.

For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.

Not requiring distinctness gives A368111, firsts of A368109, sorted A368112.

For multisets of indices we have A368184, firsts of A368183, sorted A368185.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A309326, A326031, A326702, A326753, A367771, A367906, A367907, A367908, A367909.

Sequence in context: A073008 A105199 A020791 * A368184 A086210 A085467

Adjacent sequences: A367907 A367908 A367909 * A367911 A367912 A367913

Keyword

nonn,more

Author

Gus Wiseman, Dec 16 2023

Status

approved