

A063673


Denominators of 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, ... } of approximations to Pi with increasing denominators, where each approximation is an improvement on its predecessors.


5



1, 4, 5, 6, 7, 57, 64, 71, 78, 85, 92, 99, 106, 113, 16604, 16717, 16830, 16943, 17056, 17169, 17282, 17395, 17508, 17621, 17734, 17847, 17960, 18073, 18186, 18299, 18412, 18525, 18638, 18751, 18864, 18977, 19090, 19203, 19316, 19429, 19542, 19655, 19768, 19881, 19994, 20107, 20220, 20333, 20446, 20559, 20672, 20785
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Pi = 3.1415926.... is an irrational number and can't be exactly represented by a fraction with rational numerator and denominators. The fraction 355/113 is so accurate that improves the approximation of Pi by five significant digits over the previous 333/106. To find a slightly more accurate approximation we have to go to 52163 / 16604.  Sergio Pimentel, Sep 13 2005
The approximations 22/7 and 355/113 were already known by Zu Chongzhi (5th century) and A. Metius, 1585. (Thanks to P. Curtz for this remark.)  M. F. Hasler, Apr 03 2013
The approximation 355/113 was used by S. Ramanujan in the paper "Squaring the circle" J. of the Indian Math. Soc., V, (1913) page 132 (http://en.wikisource.org/wiki/Squaring_the_circle) to give a geometrical construction of a square whose area is approximately equal to that of a circle.  Juan Monterde, Jul 26 2013


LINKS

P. D. Howard, Table of n, a(n) for n = 0..18865


EXAMPLE

333/106 = 3.1415094... is 99.99% accurate
355/113 = 3.1415929... is 99.99999% accurate


PROG

(PARI) A063673(limit) = { local(best, tmp); best=Pi3; for(n=1, limit, tmp=abs(round(Pi*n)/nPi); if(tmp<best, best=tmp; print1(n", ") ) ) } \\ Charles R Greathouse IV, Aug 23 2006
(APL (NARS2000)) B⍸∪⌊\B←(○1)(⌊.5+○A)÷A←⍳100000 \\ Michael Turniansky, Jun 09 2015


CROSSREFS

Cf. A063674, A057082.
Sequence in context: A010754 A187807 A051036 * A105737 A257816 A033597
Adjacent sequences: A063670 A063671 A063672 * A063674 A063675 A063676


KEYWORD

frac,nonn


AUTHOR

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


EXTENSIONS

More terms from Charles R Greathouse IV, Aug 23 2006, and from M. F. Hasler, Apr 03 2013


STATUS

approved



