A141823 -id:A141823 - cp's OEIS frontend

A141832 Integers n>1 such that A141822(n)=2.

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 21, 26, 27, 29, 30, 31, 34, 41, 43, 44, 45, 46, 47, 49, 50, 55, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 79, 80, 81, 89, 97, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 115, 116, 117, 119, 121, 123, 128, 129, 131, 144

Offset: 1

T. D. Noe, Jul 09 2008

nonn

Zaremba conjectured that A141823, A141833, A195901, and this sequence form a partition of the integers >1.

Is this sequence finite or infinite?

T. D. Noe, Table of n, a(n) for n=1..4398

Cf. A141821.

A141822 Maximum term in the continued fraction of A141821(n)/n.

Original entry on oeis.org

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

Offset: 2

T. D. Noe, Jul 08 2008

nonn

Consider the continued fraction [0;c1,c2,...,cm] of k/n, with k

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.

Robin Visser, Table of n, a(n) for n = 2..10000 (terms n = 2..2000 from T. D. Noe).

See A141821 for the least value of k for each n.
See A141832, A141833, A141823, and A195901 for the integers n>1 such that a(n) = 2, 3, 4, and 5, respectively.
Cf. A006839 (where cm is constrained to be 1).

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

vecmax(v)=my(mx=v[1]); for(i=2,#v,mx=max(mx,v[i])); mx
a(n)=vecmin([vecmax(contfrac(k/n))|k<-[1..n],gcd(k,n)==1]) \\ Charles R Greathouse IV, Jul 18 2014

Edited by Max Alekseyev, Sep 25 2011

A141833 Integers n>1 such that A141822(n)=3.

Original entry on oeis.org

4, 9, 10, 14, 15, 16, 22, 23, 24, 25, 32, 33, 35, 36, 37, 39, 40, 48, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 66, 72, 77, 78, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 102, 103, 107, 110, 113, 114, 118, 120, 122, 124, 125, 126, 127, 130, 132, 133, 134, 135, 136

Offset: 1

T. D. Noe, Jul 09 2008

nonn

Zaremba conjectured that A141823, A141832, A195901, and this sequence form a partition of the integers >1.Is this sequence finite or infinite?

T. D. Noe, Table of n, a(n) for n=1..5563

Cf. A141821.

A195901 Integers n>1 such that A141822(n)=5.

Original entry on oeis.org

6, 54, 150

Offset: 1

Max Alekseyev, Sep 25 2011

No other terms below 10^6.
Zaremba conjectured that A141823, A141832, A141833, and this sequence form a partition of the integers >1.
Is this sequence finite? If so, is it complete?

A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc., 50 (2013), 187-228. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A141823, A141832, A141833.

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A141832 Integers n>1 such that A141822(n)=2.

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A141822 Maximum term in the continued fraction of A141821(n)/n.

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A141833 Integers n>1 such that A141822(n)=3.

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A195901 Integers n>1 such that A141822(n)=5.

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