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.
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, 12, 25, 10, 13, 27, 11, 10, 9, 14, 11, 29, 11, 31, 31, 19, 17, 34, 37, 18, 19, 40, 41, 14, 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
Offset: 2
Keywords
Examples
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.
References
- R. K. Guy, Unsolved problems in number theory, F20.
- 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).
Links
- Robin Visser, Table of n, a(n) for n = 2..10000 (terms n = 2..2000 from T. D. Noe).
- T. W. Cusick, Zaremba's conjecture and sums of the divisor function, Math. Comp. 61 (1993), 171-176.
- Takao Komatsu, On a Zaremba's conjecture for powers, Sarajevo J. Math. 1 (2005), 9-13.
Programs
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Mathematica
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}]
Comments