A327360 Minimal numerator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

3, 44, 355, 3195, 99733, 833719, 5419351, 80143857, 657408909, 6167950454, 42106686282, 983339177173, 8958937768937, 94960529682104, 428224593349304, 6134899525417045, 66627445592888887, 430010946591069243, 5293386250278608690, 31760317501671652140

Offset

1,1

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

Jon E. Schoenfield, Magma program

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

A327361 gives the corresponding denominators.

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: A133073 A055539 A046946 * A092545 A353878 A359631

Adjacent sequences: A327357 A327358 A327359 * A327361 A327362 A327363

Keyword

base,frac,nonn

Author

Jason Zimba, Sep 03 2019

Extensions

Terms a(10) and beyond from Jon E. Schoenfield, Mar 11 2021

Status

approved