OFFSET
0,4
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 = 0..10000
John Tyler Rascoe, Log scatterplot of a(n), n=0..32906.
FORMULA
a(A072639(n)) = n. - John Tyler Rascoe, Jul 15 2024
EXAMPLE
The set-system {{1,2},{1,4},{3}} with BII-number 268 has two connected components, so a(268) = 2.
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[csm[bpe/@bpe[n]]], {n, 0, 100}]
PROG
(Python)
from sympy.utilities.iterables import connected_components
def bin_i(n): #binary indices
return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def A326753(n):
E, a = [], [bin_i(k) for k in bin_i(n)]
m = len(a)
for i in range(m):
for j in a[i]:
for k in range(m):
if j in a[k]:
E.append((i, k))
return(len(connected_components((list(range(m)), E)))) # John Tyler Rascoe, Jul 16 2024
CROSSREFS
Positions of 0's and 1's are A326749.
Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A304716, A305078, A305079 (same for MM-numbers), A323818, A326031, A326702.
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved