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.

2, 3, 4, 6, 13, 28, 64, 756, 126225, 157366665

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: A073146 A038767 A363198 * A369849 A174046 A095991

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