A327361 Minimal denominator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

1, 14, 113, 1017, 31746, 265381, 1725033, 25510582, 209259755, 1963319607, 13402974518, 313006581566, 2851718461558, 30226875395063, 136308121570117, 1952799169684491, 21208174623389167, 136876735467187340, 1684937174853026414, 10109623049118158484

Offset

1,2

References

O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255. (Provides first 8 terms. Also contains similar sequences for sqrt(2) and e.)

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000

O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255.

Example

The fractions with 2-digit numerators and 2-digit denominators that best approximate Pi are 44/14 and 88/28. The fraction with 6-digit numerator and 6-digit denominator that best approximates Pi is 833719/265381.

Mathematica

(* Given the 8th term, find the 9th term *)
(* This took twelve-plus hours to run on a laptop *)
ResultList = {};
nVal = 9;
tol = Abs[80143857/25510582 - Pi]; (* 80143857 is A327360(8), 25510582 is A327361(8) *)
Do[
  CurrentNumerator = i;
  Do[
   CurrentDenominator = j;
   CurrentQuotient = N[CurrentNumerator/CurrentDenominator];
   If[
    Abs[CurrentQuotient - Pi] <= tol,
    ResultList = Append[ResultList, {CurrentNumerator, CurrentDenominator}]
    ],
   {j, Floor[i/(Pi + tol)], Floor[i/(Pi - tol)] + 1}],
  {i, Floor[(Pi - tol)*10^(nVal - 1)], (10^nVal - 1)}];
DifferenceList =
  Table[
   Abs[ResultList[[i, 1]]/ResultList[[i, 2]] - Pi],
   {i, 1, Length[ResultList]}];
Extract[ResultList, Position[DifferenceList, Min[DifferenceList]]]

Crossrefs

A327360 gives the corresponding numerators.

Cf. A072398/A072399, which gives the best rational approximation to Pi subject to a different constraint.

Cf. A002485/A002486, A063674/A063673, A325158/A325159.

Sequence in context: A155655 A007817 A285147 * A293874 A044346 A044727

Adjacent sequences: A327358 A327359 A327360 * A327362 A327363 A327364

Keyword

base,frac,nonn

Author

Jason Zimba, Sep 03 2019

Extensions

a(10)-a(20) from Jon E. Schoenfield, Mar 12 2021

Status

approved