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

0,1

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->infinity} (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.

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

P. Diaconis and P. Erdos, 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

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

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

See A002117 for further references and links.

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

Sequence in context: A249651 A021626 A320376 * A245700 A201521 A011393

Adjacent sequences: A059953 A059954 A059955 * A059957 A059958 A059959

Keyword

easy,nonn,cons

Author

Jason Earls, Mar 01 2001

Status

approved