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%I #27 Mar 19 2021 06:59:34
%S 1,1,9,2,6,8,6,6,4,6,8,0,7,1,6,0,9,3,7,9,6,5,8,7,1,8,0,1,8,1,3,7,7,7,
%T 2,5,5,0,4,5,7,1,8,5,5,7,9,6,6,9,0,6,0,1,5,9,9,9,1,3,9,2,9,9,0,1,8,8,
%U 4,2,4,4,3,0,3,9,2,8,9,9,9,9,3,8,4,6,7,1,3,5,6,2,6,4,8,5,1,8,4,7,0,8,1,1,9,8,0,9,1,2,9,5,4,1,6,5,5,9,6,6
%N Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.
%C Decimal expansion of sum of squares of reciprocals of nonprime numbers.
%C Decimal expansion of the nonprime zeta function at 2.
%C Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]
%C More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.
%C Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...
%C Dirichlet g.f. of A005171: nonprime zeta(s).
%H G. C. Greubel, <a href="/A275647/b275647.txt">Table of n, a(n) for n = 1..5000</a>
%H Ilya Gutkovskiy, <a href="/A275647/a275647.pdf">Nonprime zeta function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html">Riemann Zeta Function 2</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.
%F Equals zeta(2) - prime zeta(2) = A013661 - A085548.
%F Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).
%F Equals Sum_{k>=1} 1/A062312(k).
%F Equals Sum_{k>=1} 1/A018252(k)^2.
%F Equals 1 + Sum_{k>=1} 1/A002808(k)^2.
%F Equals A222171 + A111003 - A085548.
%e 1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...
%t RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
%t RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]
%o (PARI) eps()=2.>>bitprecision(1.)
%o primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
%o zeta(2) - primezeta(2) \\ _Charles R Greathouse IV_, Aug 05 2016
%o (PARI) Pi^2/6 - sumeulerrat(1/p, 2) \\ _Amiram Eldar_, Mar 19 2021
%Y Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.
%K nonn,cons
%O 1,3
%A _Ilya Gutkovskiy_, Aug 04 2016
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