%I #10 Jun 14 2021 02:32:27

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

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

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

%N Decimal expansion of Product_{p primes} sqrt(1 + 1/(4*p*(p-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. 50.

%F Equals sqrt(Pi) * lim_{n->infinity} sqrt(log(n))/n * Sum_{k=1..n} 1/A034444(k).

%e 1.09698311911435715092514804937509666493621167605436728776543452986946366666...

%t $MaxExtraPrecision = 1000; Clear[f]; f[p_] := Sqrt[1 + 1/(4*p*(p-1))]; Do[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]]], {m, 100, 500, 100}]

%o (PARI) sqrt(prodeulerrat(1 + 1/(4*p*(p-1)))) \\ _Amiram Eldar_, Jun 13 2021

%Y Cf. A034444, A083281, A345231.

%K nonn,cons

%O 1,3

%A _Vaclav Kotesovec_, Jun 13 2021