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%I A367910 #5 Dec 17 2023 11:23:47
%S A367910 7,1,4,20,68,320,352,1088,3136,13376,16704,5184,82240,70720,17472
%N A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.
%C A367910 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}.
%H A367910 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A367910 The terms together with the corresponding set-systems begin:
%e A367910       7: {{1},{2},{1,2}}
%e A367910       1: {{1}}
%e A367910       4: {{1,2}}
%e A367910      20: {{1,2},{1,3}}
%e A367910      68: {{1,2},{1,2,3}}
%e A367910     320: {{1,2,3},{1,4}}
%e A367910     352: {{2,3},{1,2,3},{1,4}}
%e A367910    1088: {{1,2,3},{1,2,4}}
%e A367910    3136: {{1,2,3},{1,2,4},{3,4}}
%e A367910   13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%e A367910   16704: {{1,2,3},{1,4},{1,2,3,4}}
%e A367910    5184: {{1,2,3},{1,2,4},{1,3,4}}
%e A367910   82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}
%e A367910   70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}
%t A367910 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A367910 c=Table[Length[Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]],{n,1000}];
%t A367910 spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
%t A367910 Table[Position[c,n][[1,1]],{n,0,spnm[c]}]
%Y A367910 Positions of first appearances in A367905.
%Y A367910 The sorted version is A367911.
%Y A367910 For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
%Y A367910 Not requiring distinctness gives A368111, firsts of A368109, sorted A368112.
%Y A367910 For multisets of indices we have A368184, firsts of A368183, sorted A368185.
%Y A367910 A048793 lists binary indices, length A000120, sum A029931.
%Y A367910 A058891 counts set-systems, covering A003465, connected A323818.
%Y A367910 A070939 gives length of binary expansion.
%Y A367910 A096111 gives product of binary indices.
%Y A367910 Cf. A072639, A309326, A326031, A326702, A326753, A367771, A367906, A367907, A367908, A367909.
%K A367910 nonn,more
%O A367910 0,1
%A A367910 _Gus Wiseman_, Dec 16 2023

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