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%I #6 Jun 13 2021 10:32:36
%S 1,4,4,3,8,6,7,5,1,0,4,7,0,6,3,9,6,5,1,1,2,5,0,4,4,3,2,4,9,0,3,8,4,7,
%T 1,1,4,5,4,5,5,5,1,9,7,8,9,7,1,7,8,2,5,5,0,2,7,7,9,4,3,3,3,8,8,9,7,7,
%U 8,5,8,1,6,4,1,4,9,9,4,4,6,6,5,6,9,3,9,5,4,5,4,8,9,0,9,3,5,2,6,8,8,5,1,8,8
%N Decimal expansion of Product_{p primes} (1 - 1/p)*(1 + (1 + 1/p)*Sum_{k>=1} 1/(p^k + p^(-k-1))).
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155 (constant C1).
%F Equals lim_{n->infinity} 1/n * Sum_{k=1..n} k^2/A057660(k).
%e 1.44386751047063965112504432490384711454555197897178255027794333889778581...
%t $MaxExtraPrecision = 1000; m = 500; Do[Clear[f]; f[p_] := (1 - 1/p)*(1 + (1 + 1/p)*Sum[1/(p^j + p^(-j - 1)), {j, 1, k}]); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; Print[f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]], {k, 100, 500, 100}]
%Y Cf. A057660, A174405, A345295.
%K nonn,cons
%O 1,2
%A _Vaclav Kotesovec_, Jun 13 2021
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