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Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.
9

%I #7 Dec 18 2023 08:28:50

%S 1,4,64,20,68,52,1088,84,308,1092,116,5184,820,1108,372,5188,2868,

%T 1140,13376,884,5204,17204,1396,13380,2932,5236,275520,19252,1908,

%U 13396,17268,5492,275524,84788,3956,13428,1324096,19316,6004,275540,215860,18292,13684

%N Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

%C A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

%e The terms together with the corresponding set-systems begin:

%e 1: {{1}}

%e 4: {{1,2}}

%e 64: {{1,2,3}}

%e 20: {{1,2},{1,3}}

%e 68: {{1,2},{1,2,3}}

%e 52: {{1,2},{1,3},{2,3}}

%e 84: {{1,2},{1,3},{1,2,3}}

%e 308: {{1,2},{1,3},{2,3},{1,4}}

%e 116: {{1,2},{1,3},{2,3},{1,2,3}}

%e 820: {{1,2},{1,3},{2,3},{1,4},{2,4}}

%e 372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

%e 884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

%t nn=10000;

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

%t dd=Select[Range[nn],Max@@First/@FactorInteger[#]<=3&];

%t qq=Table[Length[Tuples[bpe/@bpe[n]]],{n,nn}];

%t kk=Select[Range[Length[dd]],SubsetQ[qq,Take[dd,#]]&]

%t Table[Position[qq,dd[[n]]][[1,1]],{n,kk}]

%Y With distinctness we have A367910, sorted A367911, firsts of A367905.

%Y For multisets we have A367913, sorted A367915, firsts of A367912.

%Y Positions of first appearances in A368109.

%Y The sorted version is A368112.

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

%Y A058891 counts set-systems, covering A003465, connected A323818.

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 17 2023