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.

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, 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, 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

Offset: 0

Author

John W. Layman, Feb 06 2002

Keywords

• nonn
• look
• nice
• frac
• hear

Comments

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

Examples

			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
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Neil Calkin and Herbert S. Wilf, Recounting the rationals, The American Mathematical Monthly, Vol. 107, No. 4 (2000), 360-363.
• Neil Calkin and Herbert S. Wilf, Recounting the rationals, Fermat's Library (2008).

Crossrefs

See A066720 for a somewhat similar sequence.

Cf. A002487, A025582, A050376, A057979.

Programs

Mathematica

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 *)

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)))
for (n=0, 88, v = if (n==0, 1, other(v)); print1 (v ", ")) \\ Rémy Sigrist, Aug 07 2017

Formula

a(6)=3, since 1/4 and 2/4 = 1/2 have already occurred as ratios of adjacent terms.