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Search: id:A141821
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| A141821 |
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Least number k<n and gcd(k,n)=1 such that the largest term of the continued fraction of k/n is as small as possible. |
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+0
5
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| 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
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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See A141822 for the value of the largest term in the continued fraction of a(n)/n. Zaremba conjectured that largest value is 5.
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REFERENCES
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T. W. Cusick, Zaremba's conjecture and sums of the divisor function, Math. Comp. 61 (1993), 171-176.
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).
R. K. Guy, Unsolved problems in number theory, F20.
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LINKS
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T. D. Noe, Table of n, a(n) for n=2..2000
Takao Komatsu, On a Zaremba's conjecture for powers, Sarajevo J. Math. 1 (2005), 9-13.
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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Sequence in context: A075274 A135737 A125179 this_sequence A144308 A144307 A144310
Adjacent sequences: A141818 A141819 A141820 this_sequence A141822 A141823 A141824
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jul 08 2008
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