A141822 - OEIS

%I #17 Nov 01 2023 13:39:27

%S 2,2,3,2,5,2,2,3,3,2,2,2,3,3,3,2,2,2,4,2,3,3,3,3,2,2,4,2,2,2,3,3,2,3,

%T 3,3,4,3,3,2,4,2,2,2,2,2,3,2,2,3,3,3,5,2,3,3,3,3,3,3,3,2,2,2,3,2,2,2,

%U 2,2,3,2,2,2,2,3,3,2,2,2,3,3,3,3,3,3,3,2,4,3,3,3,3,3,4,2,3,2,2,2,3,3,2,2,2

%N Maximum term in the continued fraction of A141821(n)/n.

%C Consider the continued fraction [0;c1,c2,...,cm] of k/n, with k<n and gcd(k,n)=1. Let f(k,n) be the maximum of the ci. Then a(n) is the minimum value of f(k,n).

%C Zaremba conjectured that a(n)<=5, a bound that is attained for n in A195901. It appears that n=150 may be the largest integer with a(n)=5, while n=6234 may be the largest integer with a(n)=4.

%H Robin Visser, <a href="/A141822/b141822.txt">Table of n, a(n) for n = 2..10000</a> (terms n = 2..2000 from T. D. Noe).

%t Table[c=ContinuedFraction[Select[Range[n-1],GCD[ #,n]==1&]/n]; Min[Max/@c], {n, 150}]

%o (PARI) vecmax(v)=my(mx=v[1]); for(i=2,#v,mx=max(mx,v[i])); mx

%o a(n)=vecmin([vecmax(contfrac(k/n))|k<-[1..n],gcd(k,n)==1]) \\ _Charles R Greathouse IV_, Jul 18 2014

%Y See A141821 for the least value of k for each n.

%Y See A141832, A141833, A141823, and A195901 for the integers n>1 such that a(n) = 2, 3, 4, and 5, respectively.

%Y Cf. A006839 (where cm is constrained to be 1).

%K nonn

%O 2,1

%A _T. D. Noe_, Jul 08 2008

%E Edited by _Max Alekseyev_, Sep 25 2011