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A327361
Minimal denominator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.
2
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.)
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.
Sequence in context: A155655 A007817 A285147 * A293874 A044346 A044727
KEYWORD
base,frac,nonn
AUTHOR
Jason Zimba, Sep 03 2019
EXTENSIONS
a(10)-a(20) from Jon E. Schoenfield, Mar 12 2021
STATUS
approved