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%I #5 Mar 30 2024 15:59:49
%S 1,2,4,8,16,32,40,64,128,160,256,288,296,416,512,520,544,552,640,672,
%T 800,808,928,1024,2048,2176,2304,2432,2560,2688,2816,2944,4096,8192,
%U 8200,8224,8232,8320,8352,8480,8488,8608,8704,8712,8736,8744,8832,8864,8992
%N Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.
%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.
%F Intersection of A087086 and A371291.
%e The terms together with their prime indices of binary indices begin:
%e 1: {{}}
%e 2: {{1}}
%e 4: {{2}}
%e 8: {{1,1}}
%e 16: {{3}}
%e 32: {{1,2}}
%e 40: {{1,1},{1,2}}
%e 64: {{4}}
%e 128: {{1,1,1}}
%e 160: {{1,2},{1,1,1}}
%e 256: {{2,2}}
%e 288: {{1,2},{2,2}}
%e 296: {{1,1},{1,2},{2,2}}
%e 416: {{1,2},{1,1,1},{2,2}}
%e 512: {{1,3}}
%e 520: {{1,1},{1,3}}
%e 544: {{1,2},{1,3}}
%e 552: {{1,1},{1,2},{1,3}}
%e 640: {{1,1,1},{1,3}}
%e 672: {{1,2},{1,1,1},{1,3}}
%e 800: {{1,2},{2,2},{1,3}}
%e 808: {{1,1},{1,2},{2,2},{1,3}}
%e 928: {{1,2},{1,1,1},{2,2},{1,3}}
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t bpe[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 Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]
%Y Connected case of A087086, relatively prime A328671.
%Y For binary indices of binary indices we have A326750, non-primitive A326749.
%Y For prime indices of prime indices we have A329559, non-primitive A305078.
%Y Primitive case of A371291 = positions of ones in A371452.
%Y For binary indices of prime indices we have A371445, non-primitive 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, A000120 length, A272020 reverse, A029931 sum.
%Y A070939 gives length of binary expansion.
%Y A096111 gives product of binary indices.
%Y A326964 counts connected set-systems, covering A323818.
%Y Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 28 2024
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