A371451 - OEIS

%I #6 Apr 01 2024 15:38:40

%S 0,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,2,1,1,2,1,2,1,2,1,1,1,1,2,1,

%T 2,2,1,2,1,1,1,3,1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,2,1,1,1,1,2,1,1,2,1,1,

%U 2,2,1,2,1,2,1,2,1,2,1,1,1,1,1,3,1,2,1

%N Number of connected components of the binary indices of the prime indices of n.

%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 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.

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

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

%t Table[Length[csm[bix/@prix[n]]],{n,100}]

%Y For prime indices of prime indices we have A305079, ones A305078.

%Y Positions of ones are A325118.

%Y Positions of first appearances are A325782.

%Y For prime indices of binary indices we have A371452, ones A371291.

%Y For binary indices of binary indices we have A326753, ones A326749.

%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, reverse A272020, length A000120, sum A029931.

%Y A070939 gives length of binary expansion.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

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

%Y Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

%K nonn

%O 1,6

%A _Gus Wiseman_, Apr 01 2024