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%I #9 Nov 16 2021 07:20:50
%S 3,5,2,2,2,1,1,0,0,4,9,9,5,8,2,7,9,6,3,6,8,3,0,1,6,7,5,1,6,3,3,1,8,6,
%T 0,5,0,9,4,2,9,3,2,1,5,7,0,8,5,4,9,0,2,4,7,0,7,1,1,3,4,2,1,2,9,5,5,4,
%U 8,5,2,8,3,4,1,0,0,3,3,8,9,2,6,8,4,0,4,1,2,2,0,7,2,8,4,6,5,6,6,8,8
%N Decimal expansion of lim_{N->infinity} (1/N^2 Sum_{n=1..N} K(n)), where K(n) is the squarefree kernel of n.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.5.1 Carefree couples, p. 111.
%F Equals (Pi^2/12)*A065464.
%F Equals (1/2) * A065463. - _Amiram Eldar_, Nov 16 2021
%e 0.35222110049958279636830167516331860509429321570854902470711342...
%t $MaxExtraPrecision = 800; digits = 101; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]]; (1/2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First
%o (PARI) prodeulerrat(1 - 1/(p*(p+1)))/2 \\ _Amiram Eldar_, Nov 16 2021
%Y Cf. A007947, A065463, A065464.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Apr 25 2016