A067992 - OEIS

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

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OFFSET

0,3

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

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).

FORMULA

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

See A066720 for a somewhat similar sequence.

Cf. A002487, A025582, A050376, A057979.

Sequence in context: A286234 A161621 A095701 * A317024 A354803 A140757

Adjacent sequences: A067989 A067990 A067991 * A067993 A067994 A067995

KEYWORD

nonn,look,nice,frac,hear

AUTHOR

John W. Layman, Feb 06 2002

STATUS

approved