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Decimal expansion of (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2.
4

%I #34 Mar 18 2021 07:35:10

%S 2,7,4,9,3,3,4,6,3,3,8,6,5,2,5,5,8,8,9,1,7,5,3,8,7,3,8,7,2,2,6,7,9,3,

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

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

%N Decimal expansion of (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2.

%C This is the probability that the gcd of any two integers is prime. - _David Cushing_, Mar 27 2013

%C The asymptotic density of integers whose largest square divisor is a square of a prime (A082293). - _Amiram Eldar_, Jul 07 2020

%H Math StackExchange, <a href="http://math.stackexchange.com/questions/39665/given-2-randomly-chosen-integers-x-y-what-is-pk-gcdx-y">Given 2 randomly chosen integers x,y what is P(k=gcd(x,y))?</a>, May 2011.

%e 0.27493346338652558891753873872267935690981646197586235178986...

%t Drop[Flatten[RealDigits[N[PrimeZetaP[2] 6/Pi^2, 100]]], -1] (* _Geoffrey Critzer_, Jan 17 2015 *)

%o (PARI) eps()=2.>>bitprecision(1.)

%o primezeta(s)=my(t=s*log(2)); sum(k=1,lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))

%o primezeta(2)*6/Pi^2 \\ _Charles R Greathouse IV_, Jul 30 2016

%o (PARI) sumeulerrat(1/p, 2)/zeta(2) \\ _Amiram Eldar_, Mar 18 2021

%Y Cf. A059956, A082293, A085548.

%K nonn,cons,nice

%O 0,1

%A _N. J. A. Sloane_, Feb 06 2013