
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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets of positive integers) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
A setsystem is an antichain if no edge is a proper subset of any other.
For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.


EXAMPLE

The sequence of terms together with their corresponding setsystems begins:
0: {}
1: {{1}}
20: {{1,2},{1,3}}
52: {{1,2},{1,3},{2,3}}
308: {{1,2},{1,3},{2,3},{1,4}}
820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
First/@GatherBy[Select[Range[0, 10000], stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&], Length[bpe[#]]&]
