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

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

Author

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

Keywords

• frac
• nonn

Comments

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

{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]

Extensions

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