This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A141821 #18 Nov 06 2023 11:08:03
%S A141821 1,2,3,2,5,5,3,7,3,8,5,5,11,4,7,12,13,7,9,8,17,7,7,7,19,19,23,12,11,
%T A141821 12,25,10,13,27,11,10,9,14,11,29,11,31,31,19,17,34,37,18,19,40,41,14,
%U A141821 17,21,15,16,17,18,47,17,23,46,45,46,25,49,49,50,29,26,19,27,31,29,55,34,61
%N A141821 Least number k < n and coprime to n such that the largest term of the continued fraction of k/n is as small as possible.
%C A141821 See A141822 for the value of the largest term in the continued fraction of a(n)/n. Zaremba conjectured that the largest value is 5.
%D A141821 R. K. Guy, Unsolved problems in number theory, F20.
%D A141821 S. K. Zaremba, ed., "Applications of number theory to numerical analysis," Proceedings of the Symposium at the Centre for Research in Mathematics, University of Montreal, Academic Press, New York, London (1972).
%H A141821 Robin Visser, <a href="/A141821/b141821.txt">Table of n, a(n) for n = 2..10000</a> (terms n = 2..2000 from T. D. Noe).
%H A141821 T. W. Cusick, <a href="https://doi.org/10.1090/S0025-5718-1993-1189517-7">Zaremba's conjecture and sums of the divisor function</a>, Math. Comp. 61 (1993), 171-176.
%H A141821 Takao Komatsu, <a href="http://www.anubih.ba/Journals/vol-1,no-1,y05/03revkomatsu.pdf">On a Zaremba's conjecture for powers</a>, Sarajevo J. Math. 1 (2005), 9-13.
%e A141821 For n=7, the six continued fractions for k/7 are (0, 7), (0, 3, 2), (0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) and (0, 1, 6). It is easy to see that the fifth one, for 5/7, has the smallest maximum term, 2. Hence a(7)=5.
%t A141821 Table[k=Select[Range[n-1], GCD[ #,n]==1&]; c=ContinuedFraction[k/n]; mx=Max/@c; mn=Min[mx]; k[[Position[mx,mn,1,1][[1,1]]]], {n,2,100}]
%K A141821 nonn
%O A141821 2,2
%A A141821 _T. D. Noe_, Jul 08 2008