OFFSET
1,3
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. 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, and {{2},{1,3}} is a set partition, it follows that 18 belongs to the sequence.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of all set partitions together with their BII numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
16: {{1,3}}
18: {{2},{1,3}}
32: {{2,3}}
33: {{1},{2,3}}
64: {{1,2,3}}
128: {{4}}
129: {{1},{4}}
130: {{2},{4}}
131: {{1},{2},{4}}
132: {{1,2},{4}}
136: {{3},{4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], UnsameQ@@Join@@bpe/@bpe[#]&]
PROG
(Python)
from itertools import chain, count, combinations, islice
from sympy.utilities.iterables import multiset_partitions
def a_gen():
yield 0
for n in count(1):
t = []
for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
if n in i:
for j in multiset_partitions(i):
t.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
yield from sorted(t)
A326701_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, May 24 2024
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 21 2019
STATUS
approved