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A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.

%I A067992 #21 Sep 03 2018 20:21:39
%S A067992 1,1,2,3,1,4,3,5,1,6,5,2,7,1,8,3,7,4,5,7,6,11,1,9,2,11,3,10,1,12,5,8,
%T A067992 7,9,4,11,5,9,8,11,7,10,9,11,10,13,1,14,3,13,2,15,1,16,3,17,1,18,5,13,
%U A067992 4,15,7,12,11,13,6,17,2,19,1,20,3,19,4,17,5,14,9,13,7,16,5,19,6,23,1,21,2
%N A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.
%C A067992 Every positive rational number appears exactly once as the ratio of adjacent terms (in either order). Conjecture: adjacent terms are always relatively prime. - _Franklin T. Adams-Watters_, Sep 13 2006
%H A067992 Rémy Sigrist, <a href="/A067992/b067992.txt">Table of n, a(n) for n = 0..10000</a>
%H A067992 Neil Calkin and Herbert S. Wilf, <a href="https://www.jstor.org/stable/2589182">Recounting the rationals</a>, The American Mathematical Monthly, Vol. 107, No. 4 (2000), 360-363.
%H A067992 Neil Calkin and Herbert S. Wilf, <a href="http://fermatslibrary.com/s/recounting-the-rationals">Recounting the rationals</a>, Fermat's Library (2008).
%F A067992 a(6)=3, since 1/4 and 2/4 = 1/2 have already occurred as ratios of adjacent terms.
%e A067992 The sequence of all rational numbers between 0 and 1 obtained by taking ratios of sorted consecutive terms begins: 1/2, 2/3, 1/3, 1/4, 3/4, 3/5, 1/5, 1/6, 5/6, 2/5, 2/7, 1/7, 1/8, 3/8, 3/7, 4/7, 4/5, 5/7, 6/7. - _Gus Wiseman_, Aug 30 2018
%t A067992 Nest[Function[seq,Append[seq,NestWhile[#+1&,1,MemberQ[Divide@@@Sort/@Partition[seq,2,1],Min[Last[seq]/#,#/Last[seq]]]&]]],{1},100] (* _Gus Wiseman_, Aug 30 2018 *)
%o A067992 (PARI) seen = Set([]); other(p) = for (v=1, oo, my (r = min(v,p)/max(v,p)); if (!set search(seen, r), seen = set union(seen, Set([r])); return (v)))
%o A067992 for (n=0, 88, v = if (n==0, 1, other(v)); print1 (v ", ")) \\ _Rémy Sigrist_, Aug 07 2017
%Y A067992 See A066720 for a somewhat similar sequence.
%Y A067992 Cf. A002487, A025582, A050376, A057979.
%K A067992 nonn,look,nice,frac,hear
%O A067992 0,3
%A A067992 _John W. Layman_, Feb 06 2002