A002485 -id:A002485 - cp's OEIS frontend

A004112 Numbers k where |cos(k)| (or |cosec(k)| or |cot(k)|) decreases monotonically to 0; also numbers k where |tan(k)| (or |sec(k)|, or |sin(k)|) increases.

0, 1, 2, 5, 8, 11, 344, 699, 1054, 1409, 1764, 2119, 2474, 2829, 3184, 3539, 3894, 4249, 4604, 4959, 5314, 5669, 6024, 6379, 6734, 7089, 7444, 7799, 8154, 8509, 8864, 9219, 9574, 9929, 10284, 10639, 10994, 11349, 11704, 12059, 12414, 12769, 13124, 13479, 13834

Offset: 1

Clark Kimberling

a(100), a(1000), and a(10000) have 5, 215, and 221 digits, respectively. - Jon E. Schoenfield, Nov 08 2019

a(n) is also the smallest nonnegative integer k such that k mod Pi is closer to Pi/2 than any previous term. - Colin Linzer, Apr 27 2022

			After the 151st term, the sequence continues 51819, 52174, 260515, 573204, 4846147, ...
|cos(4846147)| = 0.000000255689511369808141413171..., |cosec(4846147)| = 1.00000000000003268856311..., or |cot(4846147)| = 0.000000255689511369816499535901...
|tan(4846147)| = 3910993.43356970986068082..., |sec(4846147)| = 3910993.43356983770543651..., |sin(4846147)| = 0.999999999999967311436888...

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000 (first 166 terms from Robert G. Wilson v)
Carlos Hernan Lopez Zapata and Nikos Mantzakouras, Diophantine Flint-Hills series: convergence, polylogarithms and pi's bound, ResearchGate (2025). See pp. 16, 69.
Eric Weisstein's World of Mathematics, Flint Hills Series
Eric Weisstein's World of Mathematics, Secant

Cf. A046947, A046956, A002485, A024814.

A068089 Decimal expansion of 104348 / 33215.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5, 3, 9, 2, 1, 4, 2, 1, 0, 4, 4, 7, 0, 8, 7, 1, 5, 9, 4, 1, 5, 9, 2, 6, 5

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.00000001056%.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Dale, Facts about Pi
Index entries for sequences related to the number Pi
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002485, A002486, A046965, A046947, A068028, A068079.

First[RealDigits[104348/33215,10,100]] (* Paolo Xausa, Nov 07 2023 *)

More terms from Sascha Kurz, Mar 23 2002

A072398 Numerator of best approximation to Pi with denominator <= 10^n.

Original entry on oeis.org

3, 22, 22, 355, 355, 312689, 1146408, 5419351, 245850922, 2549491779, 21053343141, 21053343141, 1783366216531, 8958937768937, 139755218526789, 428224593349304, 30246273033735921, 66627445592888887

Offset: 0

Rick L. Shepherd, Jun 15 2002

			A072398(5) = 312689 because A072398(5)/A072399(5) = 312689/99532 is the best rational approximation to Pi with positive denominator <= 10^5 = 100000. This approximation is accurate to 0.00000000092766%.

Daniel Mondot, Table of n, a(n) for n = 0..999

Cf. A072399 (denominators), A000796 (Pi), A068089, A002485/A002486.

nmax = 17; cv = Convergents[Pi, 2*nmax] // Reverse; a[n_] := Select[cv, Denominator[#] <= 10^n &, 1] // Numerator // First; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 04 2013 *)

for(n=0,40,print1(numerator(bestappr(Pi,10^n)),",")) \\ Finds these approximations very quickly.

A063674 Numerators of increasingly better rational approximations to Pi with increasing denominators (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, ...)

Original entry on oeis.org

3, 13, 16, 19, 22, 179, 201, 223, 245, 267, 289, 311, 333, 355, 52163, 52518, 52873, 53228, 53583, 53938, 54293, 54648, 55003, 55358, 55713, 56068, 56423, 56778, 57133, 57488, 57843, 58198, 58553, 58908, 59263, 59618, 59973, 60328, 60683, 61038, 61393, 61748

Offset: 1

Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001

Numerators of the sequence (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 52163/16604, 52518/16717, ...)

Large jumps occur after the classical approximations 22/7 and 355/113, which are sufficiently precise to require a much larger denominator for a better approximation. - M. F. Hasler, Apr 01 2013

P. D. Howard, Table of n, a(n) for n = 1..18865
Jean-Louis Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT], 2018.

Cf. A063673 (denominators), A057082, A002485/A002486, A132049/A132050, A072398/A072399.

piapprox[n_] := Block[{a, i}, a = {3/1}; For[i = 2, i <= n, i++, If[Abs[Round[i Pi]/i - Pi] < Abs[Last[a] - Pi], AppendTo[a, Round[i Pi]/i], Null]]; Return[a]] (* Suren Fernando via Alexander R. Povolotsky, Aug 03 2008 *)

{e=1; for(d=1,1e5, abs( Pi-round(Pi*d)/d ) < e & !print1(round(Pi*d)",") & e=abs(Pi - round(Pi*d)/d))} \\ [M. F. Hasler, Apr 01 2013]

More terms from M. F. Hasler, Apr 01 2013

A072399 Denominator of best approximation to Pi with denominator <= 10^n.

Original entry on oeis.org

1, 7, 7, 113, 113, 99532, 364913, 1725033, 78256779, 811528438, 6701487259, 6701487259, 567663097408, 2851718461558, 44485467702853, 136308121570117, 9627687726852338, 21208174623389167, 842468587426513207

Offset: 0

Rick L. Shepherd, Jun 15 2002

			a(6) = 364913 because A072398(6)/a(6) = 1146408/364913 is the best rational approximation to Pi with positive denominator <= 10^6 = 1000000. This approximation is accurate to 0.000000000051271%.

Daniel Mondot, Table of n, a(n) for n = 0..999

Cf. A072398 (numerators), A000796 (Pi), A068089, A002485/A002486.

nmax = 18; cv = Convergents[Pi, 2*nmax] // Reverse; a[n_] := Select[cv, Denominator[#] <= 10^n &, 1] // Denominator // First; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 04 2013 *)

for(n=0,40,print1(denominator(bestappr(Pi,10^n)),",")) \\ Finds these approximations very quickly.

A096456 Numerators of convergents to Pi/2.

Original entry on oeis.org

1, 2, 3, 11, 344, 355, 51819, 52174, 260515, 573204, 4846147, 5419351, 37362253, 42781604, 122925461, 411557987, 534483448, 2549491779, 3083975227, 17969367914, 21053343141, 881156436695, 902209779836, 2685575996367

Offset: 1

N. J. A. Sloane, Aug 16 2004

			1, 2, 3/2, 11/7, 344/219, 355/226, ...

M. F. Hasler, Table of n, a(n) for n = 1..1500, Oct 13 2020
I. Rosenholtz, Tangent sequences, world records, ..., Math. Mag., 72 (No. 5, 1999), 367-376.

Cf. A096463 (denominators), A053300.

Cf. A002485 (numerators of convergents to Pi).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Pi/2, n]]], {n, 1, 25}] (* Stefan Steinerberger, Mar 18 2006 *)

contfracpnqn(c=contfrac(Pi/2),#c)[1,] \\ M. F. Hasler, Oct 13 2020

More terms from Stefan Steinerberger, Mar 18 2006

A355622 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(a(n)/R(a(n))-Pi) is minimized.

Original entry on oeis.org

1, 92, 581, 5471, 52861, 998713, 7774742, 93630892, 422334431, 9190135292, 45425395441, 472539314051, 5784475521481, 49371008251751, 939253175379892, 9265811239939492, 52949745472445861, 952186420153090303, 9836241210282790313, 36386277546811128511, 442327789252803797041

Offset: 1

Stefano Spezia, Jul 10 2022

a(n) and R(a(n)) have the same number of digits.

Petr Beckmann wrote that the fraction 92/29, corresponding to the second term of the sequence, appeared as value of Pi in a document written in A.D. 718.

			n              fraction    approximated value
-   -------------------    ------------------
1                     1    1
2                 92/29    3.1724137931034...
3               581/185    3.1405405405405...
4             5471/1745    3.1352435530086...
5           52861/16825    3.1418127786033...
6         998713/317899    3.1416047235128...
7       7774742/2474777    3.1415929596889...
8     93630892/29803639    3.1415926088757...
9   422334431/134433224    3.1415926690860...
...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.

Bert Dobbelaere, Table of n, a(n) for n = 1..22

Cf. A000796, A002485, A004086, A067251.

Cf. A355623 (denominator or digital reversal).

nmax=9; a={}; For[n=1, n<=nmax, n++, minim=Infinity; For[k=10^(n-1), k<=10^n-1, k++, If[(dist=Abs[k/FromDigits[Reverse[IntegerDigits[k]]]-Pi]) < minim && Last[IntegerDigits[k]]!=0 && GCD[k,FromDigits[Reverse[IntegerDigits[k]]]]==1, minim=dist; kmin=k]]; AppendTo[a, kmin]]; a

a(10)-a(19) from Bert Dobbelaere, Jul 17 2022

a(20)-a(21) from Bert Dobbelaere, Sep 05 2022

A355623 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(R(a(n))/a(n)-Pi) is minimized.

Original entry on oeis.org

1, 29, 185, 1745, 16825, 317899, 2474777, 29803639, 134433224, 2925310919, 14459352454, 150413935274, 1841255744875, 15715280017394, 298973571352939, 2949399321185629, 16854427454794925, 303090351024681259, 3130972820121426389, 11582111864577268363, 140797308252987723244

Offset: 1

Stefano Spezia, Jul 10 2022

a(n) and R(a(n)) have the same number of digits.

Petr Beckmann wrote that the fraction 92/29, corresponding to the second term of the sequence, appeared as value of Pi in a document written in A.D. 718.

			n              fraction    approximated value
-   -------------------    ------------------
1                     1    1
2                 92/29    3.1724137931034...
3               581/185    3.1405405405405...
4             5471/1745    3.1352435530086...
5           52861/16825    3.1418127786033...
6         998713/317899    3.1416047235128...
7       7774742/2474777    3.1415929596889...
8     93630892/29803639    3.1415926088757...
9   422334431/134433224    3.1415926690860...
...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.

Bert Dobbelaere, Table of n, a(n) for n = 1..22

Cf. A000796, A002485, A004086, A067251.

Cf. A355622 (numerator or digital reversal).

nmax=9; a={}; For[n=1, n<=nmax, n++, minim=Infinity; For[k=10^(n-1), k<=10^n-1, k++, If[(dist=Abs[FromDigits[Reverse[IntegerDigits[k]]]/k-Pi]) < minim && Last[IntegerDigits[k]]!=0 && GCD[k,FromDigits[Reverse[IntegerDigits[k]]]]==1, minim=dist; kmin=k]]; AppendTo[a, kmin]]; a

a(10)-a(19) from Bert Dobbelaere, Jul 17 2022

a(20)-a(21) from Bert Dobbelaere, Sep 05 2022

A265735 Integers in the interval [Pik - 1/k, Pik + 1/k] for some k > 0.

Original entry on oeis.org

3, 4, 6, 19, 22, 44, 66, 88, 333, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 103993, 104348, 208341, 312689, 521030, 833719, 1146408, 2292816, 4272943, 5419351, 10838702, 16258053, 80143857, 85563208

Offset: 1

Michel Lagneau, Dec 15 2015

nonn

Conjecture: the sequence is infinite.
See the reference for a similar problem with Fibonacci numbers.
For k > 1, the interval [Pi*k - 1/k, Pi*k + 1/k] contains exactly one integer.
The corresponding integers k are 1, 2, 6, 7, 14, 21, 28,...(see A265739).
We observe two properties:
(1) a(n) = m*a(n-m+1) for some n, m=2,3,4.
Examples:
m = 2 => a(7)=2*a(6), a(11)=2*a(10), a(15)=2*a(14), a(20)=2*a(19), a(25)=2*a(24), a(30)=2*a(29),...
m = 3 => a(16)=3*a(14), a(21)=3*a(19), a(26)=3*a(24), a(31)=3*a(29),...
m = 4 => a(4)=4*a(1), a(32)=4*a(29), ...
But, for m=5, the formula (1) is not valid. We find a(13)=5*a(9), a(18)=5*a(10), a(23)=5*a(11), ...
(2) a(n+2) = a(n) + a(n+1) for n = 4, 9, 26, 27, 28, 29, 35, ...
For k > 1, the integer satisfying the definition is such that ceiling(Pi*k - 1/k) = floor(Pi*k + 1/k). - Stefano Spezia, Apr 26 2023

			For k=1 there exists two integers a(1)=3 and a(2)=4 in the interval [1*Pi -1/1, 1*Pi + 1/1] = [2.14159...,4.14159...];
for k=2, the number a(3)=6 is in the interval [2*Pi-1/2, 2*Pi+1/2] = [5.783185..., 6.783185...];
for k=6, the number a(4)= 19 is in the interval [6*Pi-1/6, 6*Pi+1/6] = [18.682889..., 19.016223...].

Stefano Spezia, Table of n, a(n) for n = 1..52
Takao Komatsu, The interval associated with a Fibonacci number, The Fibonacci Quarterly, Volume 41, Number 1, February 2003.
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A000796, A265739. Contains A002485 (without the first two terms) as a subsequence.

*** the program gives the interval [a,b],a(n) and k ***
nn:=10^9:
for n from 1 to nn do:
x1:=evalhf(Pi*n-1/n):y1:=evalhf(Pi*n+1/n):
x:=floor(x1):y:=floor(y1):
for j from x+1 to y do:
printf("%g %g %d %d\n",x1,y1,j,n):
od:
od:

kmax=10^9; Flatten[Table[Range[Ceiling[Pi k-1/k], Floor[Pi k+1/k]], {k, kmax}]] (* or limiting memory usage *)
a = {3,4}; kmax = 10^9; For[k = 1, k <= kmax, k++,
 If[(nw = Ceiling[Pi k - 1/k]) == Floor[Pi k + 1/k],
  AppendTo[a, nw]]]; a (* Stefano Spezia, Apr 26 2023 *)

A325158 Numerators of convergents to Pi using best rational approximation whose denominator is between consecutive powers of 2: [2^n, 2^(n+1)-1], where n = 0, 1, 2, ...

Original entry on oeis.org

3, 6, 22, 44, 66, 179, 355, 710, 1065, 1775, 3550, 6745, 13135, 25915, 102928, 104348, 312689, 625378, 1146408, 3126535, 5419351, 10838702, 16258053, 47627751, 80143857, 165707065, 411557987, 657408909, 1068966896, 2549491779, 6167950454, 12335900908, 21053343141, 42106686282

Offset: 0

Serguei Zolotov, Apr 04 2019

nonn

			The convergents are 3/1, 6/2, 22/7, 44/14, 66/21, 179/57, 355/113, 710/226, 1065/339, 1775/565, 3550/1130, 6745/2147, 13135/4181, 25915/8249, 102928/32763, ... = A325158/A325159.

Serguei Zolotov, Table of n, a(n) for n = 0..1023
E. Charrier and L. Buzer, Approximating a real number by a rational number with a limited denominator: A geometric approach, Discrete Applied Mathematics, Volume 157, Issue 16, 28 August 2009, Pages 3473-3484.
Serguei Zolotov, Python program to calculate b-file

Cf. A325159 (denominators), A002485.

cp's OEIS Frontend

A004112 Numbers k where |cos(k)| (or |cosec(k)| or |cot(k)|) decreases monotonically to 0; also numbers k where |tan(k)| (or |sec(k)|, or |sin(k)|) increases.

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A068089 Decimal expansion of 104348 / 33215.

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A072398 Numerator of best approximation to Pi with denominator <= 10^n.

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A063674 Numerators of increasingly better rational approximations to Pi with increasing denominators (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, ...)

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A072399 Denominator of best approximation to Pi with denominator <= 10^n.

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A096456 Numerators of convergents to Pi/2.

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A355622 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(a(n)/R(a(n))-Pi) is minimized.

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A355623 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(R(a(n))/a(n)-Pi) is minimized.

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A265735 Integers in the interval [Pik - 1/k, Pik + 1/k] for some k > 0.