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%I #81 Sep 08 2022 08:45:03
%S 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,
%T 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,
%U 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
%N Decimal expansion of 6/Pi^2.
%C "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.
%C 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]
%C 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
%C 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
%C 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
%C 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
%D Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333.
%D C. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 359.
%H Harry J. Smith, <a href="/A059956/b059956.txt">Table of n, a(n) for n = 0..20000</a>
%H P. Diaconis and P. Erdos, <a href="http://dx.doi.org/10.1214/lnms/1196285379">On the distribution of the greatest common divisor</a>, in A Festschrift for Herman Rubin, pp. 56-61, IMS Lecture Notes Monogr. Ser., 45, Inst. Math. Statist., Beachwood, OH, 2004
%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a>
%H H. J. Smith, <a href="http://harry-j-smith-memorial.com/download.html#XPCalc">XPCalc</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html">Hafner-Sarnak-McCurley Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RelativelyPrime.html">Relatively Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 1/A013661.
%F 6/Pi^2 = Product_{k>=1} (1 - 1/prime(k)^2) = Sum_{k>=1} mu(k)/k^2. - _Vladeta Jovovic_, May 18 2001
%e .6079271018540266286632767792583658334261526480...
%p evalf(1/Zeta(2)) ; # _R. J. Mathar_, Mar 27 2013
%t RealDigits[ 6/Pi^2, 10, 105][[1]]
%t RealDigits[1/Zeta[2], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 20 2017 *)
%o (Harry J. Smith's VPcalc program): 150 M P x=6/Pi^2.
%o (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
%o (Magma) R:= RealField(100); 6/(Pi(R))^2; // _G. C. Greubel_, Mar 09 2018
%Y See A002117 for further references and links.
%Y Cf. A005117 (squarefree numbers), A013661, A082695.
%K easy,nonn,cons
%O 0,1
%A _Jason Earls_, Mar 01 2001
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