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%I #6 Apr 01 2024 15:38:34
%S 1,1,2,1,2,2,3,1,2,1,2,2,3,2,3,1,2,2,3,2,3,3,4,2,3,2,3,3,4,3,4,1,2,1,
%T 2,1,2,1,2,1,2,1,2,1,2,1,2,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,1,2,2,3,2,
%U 3,3,4,2,3,2,3,3,4,3,4,2,3,3,4,3,4,4,5
%N Number of connected components of the prime indices of the binary indices of n.
%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.
%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.
%e The prime indices of binary indices of 281492156579880 are {{1,1},{1,2},{3,4},{4,4}}, with 2 connected components {{1,1},{1,2}} and {{3,4},{4,4}}, so a(281492156579880) = 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[prix/@bix[n]]],{n,100}]
%Y Positions of first appearances are A080355, opposite A325782.
%Y For prime indices of prime indices we have A305079, ones A305078.
%Y For binary indices of binary indices we have A326753, ones A326749.
%Y Positions of ones are A371291.
%Y For binary indices of prime indices we have A371451, ones A325118.
%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 1,3
%A _Gus Wiseman_, Apr 01 2024
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