OFFSET
1,2
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.
The empty set is not considered connected.
EXAMPLE
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
8: 1000 ~ {4}
10: 1010 ~ {2,4}
16: 10000 ~ {5}
32: 100000 ~ {6}
34: 100010 ~ {2,6}
36: 100100 ~ {3,6}
38: 100110 ~ {2,3,6}
40: 101000 ~ {4,6}
42: 101010 ~ {2,4,6}
44: 101100 ~ {3,4,6}
46: 101110 ~ {2,3,4,6}
64: 1000000 ~ {7}
128: 10000000 ~ {8}
130: 10000010 ~ {2,8}
136: 10001000 ~ {4,8}
138: 10001010 ~ {2,4,8}
160: 10100000 ~ {6,8}
162: 10100010 ~ {2,6,8}
164: 10100100 ~ {3,6,8}
MATHEMATICA
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]]]]]]]]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], Length[csm[prix/@bpe[#]]]==1&]
CROSSREFS
For prime indices of each prime index we have A305078.
The opposite version is A325118.
For binary indices of each binary index we have A326749.
Positions of ones in A371452.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Mar 27 2024
STATUS
approved