OFFSET
0,1
COMMENTS
"6/Pi^2 is the probability that two randomly selected numbers will be coprime and also the probability that a randomly selected integer is 'squarefree.'" [Hardy and Wright] - C. Pickover.
In fact, the probability that any k randomly selected numbers will be coprimes is 1/Sum_{n>=1} n^(-k) = 1/zeta(k). - Robert G. Wilson v [corrected by Ilya Gutkovskiy, Aug 18 2018]
6/Pi^2 is also the diameter of a circle whose circumference equals the ratio of volume of a cuboid to the inscribed ellipsoid. 6/Pi^2 is also the diameter of a circle whose circumference equals the ratio of surface area of a cube to the inscribed sphere. - Omar E. Pol, Oct 08 2011
6/(Pi^2 * n^2) is the probability that two randomly selected positive integers will have a greatest common divisor equal to n, n >= 1. - Geoffrey Critzer, May 28 2013
Equals lim_{n->oo} (Sum_{k=1..n} phi(k)/k)/n, i.e., the limit mean value of phi(k)/k, where phi(k) is Euler's totient function. Proof is trivial using the formula for Sum_{k=1..n} phi(k)/k listed at the Wikipedia link. For the limit mean value of k/phi(k), see A082695. - Stanislav Sykora, Nov 14 2014
This is the probability that a random point on a square lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020
REFERENCES
Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333.
C. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 359.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 28.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..20000
Persi Diaconis and Paul Erdős, On the distribution of the greatest common divisor, in A Festschrift for Herman Rubin, pp. 56-61, IMS Lecture Notes Monogr. Ser., 45, Inst. Math. Statist., Beachwood, OH, 2004.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
H. J. Smith, XPCalc. [Broken link]
Eric Weisstein's World of Mathematics, Hafner-Sarnak-McCurley Constant.
Eric Weisstein's World of Mathematics, Relatively Prime.
Eric Weisstein's World of Mathematics, Squarefree.
Wikipedia, Euler's totient function.
FORMULA
Equals 1/A013661.
6/Pi^2 = Product_{k>=1} (1 - 1/prime(k)^2) = Sum_{k>=1} mu(k)/k^2. - Vladeta Jovovic, May 18 2001
EXAMPLE
.6079271018540266286632767792583658334261526480...
MAPLE
evalf(1/Zeta(2)) ; # R. J. Mathar, Mar 27 2013
MATHEMATICA
RealDigits[ 6/Pi^2, 10, 105][[1]]
RealDigits[1/Zeta[2], 10, 111][[1]] (* Robert G. Wilson v, Jan 20 2017 *)
PROG
(Harry J. Smith's VPcalc program): 150 M P x=6/Pi^2.
(PARI) default(realprecision, 20080); x=60/Pi^2; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b059956.txt", n, " ", d)); \\ Harry J. Smith, Jun 30 2009
(Magma) R:= RealField(100); 6/(Pi(R))^2; // G. C. Greubel, Mar 09 2018
CROSSREFS
KEYWORD
AUTHOR
Jason Earls, Mar 01 2001
STATUS
approved