A059956 Decimal expansion of 6/Pi^2.

6, 0, 7, 9, 2, 7, 1, 0, 1, 8, 5, 4, 0, 2, 6, 6, 2, 8, 6, 6, 3, 2, 7, 6, 7, 7, 9, 2, 5, 8, 3, 6, 5, 8, 3, 3, 4, 2, 6, 1, 5, 2, 6, 4, 8, 0, 3, 3, 4, 7, 9, 2, 9, 3, 0, 7, 3, 6, 5, 4, 1, 9, 1, 3, 6, 5, 0, 3, 8, 7, 2, 5, 7, 7, 3, 4, 1, 2, 6, 4, 7, 1, 4, 7, 2, 5, 5, 6, 4, 3, 5, 5, 3, 7, 3, 1, 0, 2, 5, 6, 8, 1, 7, 3, 3

Offset: 0

Author

Jason Earls, Mar 01 2001

Keywords

• easy
• nonn
• cons

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

Examples

			.6079271018540266286632767792583658334261526480...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4, p. 18.
• 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.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 118-119.
• 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.
• Harry 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.
• Index entries for transcendental numbers.

Crossrefs

See A002117 for further references and links.

Cf. A005117 (squarefree numbers), A013661, A082695.

Programs

Magma

R:= RealField(100); 6/(Pi(R))^2; // G. C. Greubel, Mar 09 2018

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 *)

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

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