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%I #23 Oct 18 2018 03:33:55
%S 3,13,16,19,22,179,201,223,245,267,289,311,333,355,52163,52518,52873,
%T 53228,53583,53938,54293,54648,55003,55358,55713,56068,56423,56778,
%U 57133,57488,57843,58198,58553,58908,59263,59618,59973,60328,60683,61038,61393,61748
%N Numerators of increasingly better rational approximations to Pi with increasing denominators (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, ...)
%C 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, ...)
%C 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
%H P. D. Howard, <a href="/A063674/b063674.txt">Table of n, a(n) for n = 1..18865</a>
%H Jean-Louis Sikorav, <a href="https://arxiv.org/abs/1807.06284">Best rational approximations of an irrational number</a>, arXiv:1807.06284 [math.NT], 2018.
%t 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 *)
%o (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]
%Y Cf. A063673 (denominators), A057082, A002485/A002486, A132049/A132050, A072398/A072399.
%K frac,nonn
%O 1,1
%A Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001
%E More terms from _M. F. Hasler_, Apr 01 2013
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