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A141822
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Maximum term in the continued fraction of A141821(n)/n.
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7
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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, 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, 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
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OFFSET
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2,1
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COMMENTS
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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).
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.
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LINKS
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MATHEMATICA
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Table[c=ContinuedFraction[Select[Range[n-1], GCD[ #, n]==1&]/n]; Min[Max/@c], {n, 150}]
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PROG
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(PARI) vecmax(v)=my(mx=v[1]); for(i=2, #v, mx=max(mx, v[i])); mx
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CROSSREFS
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See A141821 for the least value of k for each n.
Cf. A006839 (where cm is constrained to be 1).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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