A371291 Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.

1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548

Offset: 1

Author

Gus Wiseman, Mar 27 2024

Keywords

• nonn
• base

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.

Examples

			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}
		

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.

The pairwise indivisible case is A371294, opposite A371445.

Positions of ones in A371452.

A001187 counts connected graphs.

A007718 counts non-isomorphic connected multiset partitions.

A048143 counts connected antichains of sets.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A087086 lists numbers whose binary indices are pairwise indivisible.

A096111 gives product of binary indices.

A326964 counts connected set-systems, covering A323818.

Cf. A000040, A000720, A001222, A305079, A326753, A371446.

Programs

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&]