A002486 -id:A002486 - cp's OEIS frontend

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

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.

A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 1.

Original entry on oeis.org

1, 2, 3, 8, 99, 33102, 66317, 265381, 1360120, 25510582, 78256779, 156513558, 209259755, 340262731, 1963319607, 6701487259, 8664806866, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 46910410813

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

			a(4) = 8 since frac(1x*) + frac(2*x) + frac(3*x) + frac(8*x) < 1, while frac(1*x) + frac(2*x) + frac(3*x) + frac(k*x) > 1 for all k > 3 and k < 8.

Cf. A002486 (denominators of convergents to Pi), A079934, A079937, A079939.

Digits := 100: a := []: s := 0: x := Pi: for n from 1 to 10000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

first(n)=my(v=vector(n),s=1.,p=Pi-3,k); for(m=1,oo, my(t=frac(p*m)); if(tCharles R Greathouse IV, Jul 25 2024

a(9) from Mark Hudson, Jan 30 2003

a(10)-a(23) from Charles R Greathouse IV, Jul 26 2024

A360367 a(n) is the denominator of the rational number with the smallest denominator that lies within 1/10^n of Pi.

Original entry on oeis.org

1, 7, 7, 64, 106, 113, 113, 24175, 32085, 33102, 99532, 265381, 1360120, 1725033, 18610450, 25510582, 78256779, 340262731, 811528438, 1963319607, 6701487259, 6701487259, 413528890451, 554260122890, 1142027682075, 2851718461558, 2851718461558, 41633749241295, 91822653867264

Offset: 0

Stefano Spezia, Feb 04 2023

			The rational numbers are 3, 22/7, 22/7, 201/64, 333/106, 355/113, 355/113, 75948/24175, ...

Wolfram Research (1988), Rationalize, Wolfram Language function, (updated 1999).

Cf. A000796, A002486, A011557.

Cf. A360366 (numerator), A360368, A360369, A360370.

Table[Denominator[Rationalize[Pi,10^(-n)]],{n,0,28}]

A156020 Denominators in an infinite sum for Pi.

Original entry on oeis.org

1, 106, 877203, 2195225334, 17599271777, 360950005720, 17348726394920, 1996375977735378, 26627865341803449, 668044491303666717, 13157161331655387213, 7653283960850915182425, 3256741424583567733172850, 388712386741794886666062286, 266182386623377135274423955447

Offset: 1

Gary W. Adamson and Alexander R. Povolotsky, Feb 01 2009

For k >= 0, define Q(k) = A002485(2k)/A002486(2k) (convergents to Pi that are less than Pi), so Pi = Sum_{k>=1} (Q(k) - Q(k-1)). Then a(n) is the denominator of Q(n) - Q(n-1).

			a(2) = 106 since A002485(4)/A002486(4) = 333/106, A002485(2)/A002486(2) = 3/1, and 333/106 - 3/1 = 15/106 (see table below).
Pi = 3/1 + 15/106 + 73/877203 + 1/2195225334 + 2/17599271777 + 3/360950005720 + 7/17348726394920 + ....
.
  n  Q(n) = A002485(2n)/A002486(2n)  Q(n) - Q(n-1)    a(n)
  -  ------------------------------  -------------  ------
  0       0/1     = 0                     -              -
  1       3/1     = 3                    3/1             1
  2     333/106   = 3.1415094339...     15/106         106
  3  103993/33102 = 3.1415926530...     73/877203   877203

Cf. A002485, A002486, A156019 (numerators).

cfPi=contfrac(Pi);
vA002485 = concat(1, contfracpnqn(cfPi, #cfPi)[1, ]);
A002485(n) = vA002485[n];
A002486(n) = contfracpnqn(vecextract(cfPi, 2^n-1))[2, 2];
a(n) = if (n==1, 1, denominator(A002485(2*n)/A002486(2*n) - A002485(2*n-2)/A002486(2*n-2))); \\ Michel Marcus, Jan 05 2022

a(n) = denominator(A002485(2n)/A002486(2n) - A002485(2n-2)/A002486(2n-2)).

More terms from Alexander R. Povolotsky, Sep 01 2009

More terms from Michel Marcus, Jan 05 2022

A156618 Denominators of Egyptian fraction for Pi-3 whose partial sums are the convergents.

Original entry on oeis.org

7, -742, 11978, -3740526, 1099482930, -2202719155, 6600663644, -26413901692, 96840976853, -496325469560, 2346251883960, -44006595799206, 1345586183756654, -4127747481719463, 10251870941174304

Offset: 0

Jaume Oliver Lafont, Feb 11 2009

sign

Numerators are all 1.

			3+1/a(0)=22/7
3+1/a(0)+1/a(1)=333/106
3+1/a(0)+1/a(1)+1/a(2)=355/113

Cf. A001466, A014013, A156019, A156020, A002485, A002486.

c0=3; for (k=2,30,m=contfracpnqn(contfrac(Pi,k));c1=m[1,1]/m[2,1];print1(1/(c1-c0),", ");c0=c1;)

A036417 Values of k for which there are no empty intervals when fractional part(m*Pi) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

Original entry on oeis.org

1, 6, 7, 106, 112, 113, 33102, 33215, 66317, 99532, 165849, 265381, 364913

Offset: 1

Eric W. Weisstein

Appears to include all the denominators of the convergents of Pi A002486. - Eric W. Weisstein, Apr 18 2024

No other terms with n <= 10^6. - Eric W. Weisstein, Apr 27 2024

Eric Weisstein's World of Mathematics, Equidistributed Sequence.
Eric Weisstein's World of Mathematics, Pi.

Cf. A036416.

Cf. A002486 (denominators of the convergents of Pi).

With[{f = FractionalPart[Pi Range[1000]]}, Position[Table[Count[BinCounts[Take[f, n], {0., 1, 1/n}], 0], {n, Length[f]}], 0]] // Flatten (* Eric W. Weisstein, Apr 27 2024 *)

a(9)-a(10) from Sean A. Irvine, Oct 31 2020

a(11)-a(13) from Eric W. Weisstein, Apr 18-19 2024

A120701 Number of unit circles which fit touching a circle of radius n-1, i.e., with their centers on a circle of radius n.

Original entry on oeis.org

2, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125, 128, 131, 135, 138, 141, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 182, 185, 188

Offset: 1

Martin Fuller, Jun 28 2006

Coincides with A022844 = floor(n*Pi) except at n=1, 25510582, ... (sequence A120702).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001116, A002486, A022844, A120702.

R:= RealField(30); [Floor(Pi(R)/Arcsin(1/n)) : n in [1..70]]; // G. C. Greubel, Aug 25 2023

Table[Floor[Pi/ArcSin[1/n]], {n, 60}] (* Indranil Ghosh, Jul 21 2017 *)

from mpmath import mp, pi, asin
mp.dps=100
def a(n): return int(floor(pi/asin(1./n)))
print([a(n) for n in range(1, 61)]) # Indranil Ghosh, Jul 21 2017

[floor(pi/arcsin(1/n)) for n in range(1,71)] # G. C. Greubel, Aug 25 2023

a(n) = floor(Pi/arcsin(1/n)).

A120702 Values of n where A022844(n) = floor(n*Pi) differs from A120701(n) = floor(Pi/arcsin(1/n)).

Original entry on oeis.org

1, 25510582, 78256779, 340262731, 1963319607, 6701487259, 1142027682075, 2851718461558, 136308121570117, 1952799169684491, 21208174623389167, 842468587426513207, 84383735478118508040, 589001211171976529866

Offset: 1

Martin Fuller, Jun 28 2006

nonn

Subsequence of A002486.

Cf. A022844, A120701, A002486.

default(realprecision,1000); isok(n) = floor(n*Pi) != floor(Pi/asin(1/n)); \\ Michel Marcus, Jan 17 2018

More terms from Max Alekseyev, Feb 22 2012

cp's OEIS Frontend

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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A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 1.

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A360367 a(n) is the denominator of the rational number with the smallest denominator that lies within 1/10^n of Pi.

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Mathematica

A156020 Denominators in an infinite sum for Pi.

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PARI

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A156618 Denominators of Egyptian fraction for Pi-3 whose partial sums are the convergents.

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PARI

A036417 Values of k for which there are no empty intervals when fractional part(m*Pi) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

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