A371291 - OEIS

%I #9 Mar 28 2024 11:58:10

%S 1,2,4,8,10,16,32,34,36,38,40,42,44,46,64,128,130,136,138,160,162,164,

%T 166,168,170,172,174,256,260,288,290,292,294,296,298,300,302,416,418,

%U 420,422,424,426,428,430,512,514,520,522,528,530,536,538,544,546,548

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

%C 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.

%C The empty set is not considered connected.

%e The terms together with their binary expansions and binary indices begin:

%e     1:          1 ~ {1}

%e     2:         10 ~ {2}

%e     4:        100 ~ {3}

%e     8:       1000 ~ {4}

%e    10:       1010 ~ {2,4}

%e    16:      10000 ~ {5}

%e    32:     100000 ~ {6}

%e    34:     100010 ~ {2,6}

%e    36:     100100 ~ {3,6}

%e    38:     100110 ~ {2,3,6}

%e    40:     101000 ~ {4,6}

%e    42:     101010 ~ {2,4,6}

%e    44:     101100 ~ {3,4,6}

%e    46:     101110 ~ {2,3,4,6}

%e    64:    1000000 ~ {7}

%e   128:   10000000 ~ {8}

%e   130:   10000010 ~ {2,8}

%e   136:   10001000 ~ {4,8}

%e   138:   10001010 ~ {2,4,8}

%e   160:   10100000 ~ {6,8}

%e   162:   10100010 ~ {2,6,8}

%e   164:   10100100 ~ {3,6,8}

%t 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]]]]]]]]];

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[0,1000],Length[csm[prix/@bpe[#]]]==1&]

%Y For prime indices of each prime index we have A305078.

%Y The opposite version is A325118.

%Y For binary indices of each binary index we have A326749.

%Y The pairwise indivisible case is A371294, opposite A371445.

%Y Positions of ones in A371452.

%Y A001187 counts connected graphs.

%Y A007718 counts non-isomorphic connected multiset partitions.

%Y A048143 counts connected antichains of sets.

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

%Y A070939 gives length of binary expansion.

%Y A087086 lists numbers whose binary indices are pairwise indivisible.

%Y A096111 gives product of binary indices.

%Y A326964 counts connected set-systems, covering A323818.

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

%K nonn,base

%O 1,2

%A _Gus Wiseman_, Mar 27 2024