A141822 -id:A141822 - 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.

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.

A141823 Integers n>1 such that A141822(n)=4.

Original entry on oeis.org

20, 28, 38, 42, 90, 96, 156, 164, 216, 228, 252, 318, 336, 350, 384, 386, 442, 508, 558, 770, 876, 922, 978, 1014, 1155, 1170, 1410, 1450, 1692, 1870, 2052, 2370, 3618, 5052, 6234

Offset: 1

T. D. Noe, Jul 08 2008, Jul 09 2008

No other terms below 10^5.
Zaremba conjectured that A141832, A141833, A195901, and this sequence form a partition of the integers >1.
Is this sequence finite? If so, is it complete?

Cf. A141832, A141833, A141821.

Edited by Max Alekseyev, Sep 25 2011

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.

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.

Original entry on oeis.org

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

T. D. Noe, Jul 08 2008

nonn

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.

			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.

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).

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.

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}]

A006839 Minimum of largest partial quotient of continued fraction for k/n, (k,n) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 1, 3, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 3, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 1, 3, 3, 2, 4, 3, 2, 2, 4, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 5, 1, 2, 3, 2, 3, 3, 2, 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, 1, 3, 2, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2

Offset: 1

N. J. A. Sloane

nonn

Consider the continued fraction [0,c_1,c_2,...,c_m] of k/n, with kA141822 only in the requirement that c_m=1. - Sean A. Irvine, Aug 12 2017

Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robin Visser, Table of n, a(n) for n = 1..10000
H. Niederreiter, Dyadic fractions with small partial quotients, Monat. f. Math., 101 (1986), 309-315.

Cf. A141822.

More terms from David W. Wilson

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

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

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A141823 Integers n>1 such that A141822(n)=4.

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

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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.

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A006839 Minimum of largest partial quotient of continued fraction for k/n, (k,n) = 1.

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