OFFSET
1,3
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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (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. Elements of a set-system are sometimes called edges.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of all covering set-systems together with their BII-numbers begins:
0: {}
1: {{1}}
3: {{1},{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
15: {{1},{2},{1,2},{3}}
18: {{2},{1,3}}
19: {{1},{2},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
26: {{2},{3},{1,3}}
27: {{1},{2},{3},{1,3}}
28: {{1,2},{3},{1,3}}
29: {{1},{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[0, 100], normQ[Join@@bpe/@bpe[#]]&]
PROG
(Python)
from itertools import chain, count, islice
def bin_i(n): #binary indices
return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
for n in count(0):
s = set(i for i in chain.from_iterable([bin_i(k) for k in bin_i(n)]))
y = len(s)
if sum(s) == (y*(y+1))//2:
yield n
A326754_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved