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A188715 Minimal largest k in set of n fractions of the form (k-1)/k all of whose ratios (smaller fraction / larger fraction) are also of that form. 0
2, 3, 4, 6, 13, 28, 64, 756, 126225, 157366665 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

These ratios, together with 1, can be the constant speeds of cyclists going forever around a circular track while only allowed to pass each other at a single point.

For all n, a(n+1)<a(n)^(2n), so the series is infinite. [John Tromp, Apr 13 2011]

LINKS

Table of n, a(n) for n=1..10.

EXAMPLE

All pairwise ratios in the set {5/6,4/5,3/4,2/3} are also of the form (k-1)/k, while for the only "lesser" size-4 set {4/5,3/4,2/3,1/2}, we have (1/2)/(4/5)=5/8 is not of that form. So a(4)=6.

PROG

(Haskell)

let ext (c, l) = [(tails.filter(\b->a*(a-1)`mod`(b-a)==0)$r, a:l) | (a:r)<-c] in map (last.snd.head) . tail . iterate (>>= ext) $ [(map reverse (inits[2..]), [])]

-- for greater efficiency, replace map reverse (inits[2..]) by

-- [m:[m-d|d<-divisors(m*(m-1)), d<m-1]|m<-[2..]], defining divisors appropriately.

CROSSREFS

Sequence in context: A066463 A073146 A038767 * A174046 A095991 A293714

Adjacent sequences:  A188712 A188713 A188714 * A188716 A188717 A188718

KEYWORD

nonn,more

AUTHOR

John Tromp, Apr 08 2011

EXTENSIONS

Finally found a(11); about the square of a(10). I doubt if a(12) will ever be found.

STATUS

approved

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Last modified May 15 23:24 EDT 2021. Contains 343937 sequences. (Running on oeis4.)