OFFSET
1,2
COMMENTS
Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice.
A binary index of k (row k of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number k to be obtained by taking the binary indices of each binary index of k. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Wikipedia, Axiom of choice.
EXAMPLE
The set-system {{2,3},{1,2,3},{1,4}} with BII-number 352 has choices such as (2,1,4) that satisfy the axiom, so 352 is in the sequence.
The terms together with the corresponding set-systems begin:
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
16: {{1,3}}
17: {{1},{1,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]!={}&]
PROG
(Python)
from itertools import count, islice, product
def bin_i(n): #binary indices
return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
for n in count(1):
for j in list(product(*[bin_i(k) for k in bin_i(n)])):
if len(set(j)) == len(j):
yield(n); break
A367906_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Dec 23 2023
CROSSREFS
The complement is A367907.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
KEYWORD
base,nonn
AUTHOR
Gus Wiseman, Dec 11 2023
STATUS
approved