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A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.

%I A360895 #25 Apr 09 2023 02:38:35
%S A360895 1,7,5,3,2,2,9,4,4,3,4,9,5,6,9,4,5,8,2,2,9,7,3,6,5,4,2,9,6,4,4,0,6,1,
%T A360895 2,8,7,6,0,5,7,4,5,8,0,2,0,2,0,7,5,4,4,5,6,1,9,0,2,9,5,1,5,6,3,1,5,3,
%U A360895 9,8,8,9,4,0,8,7,8,0,7,2,0,6,0,7,2,4,5,3,1,0,5,5,5,8,8,8,6,7,4,0,5,2,0,2,4,3,4,3,7,6,8,4,6,4,2
%N A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.
%C A360895 Theorem: Let p(n) be the smallest prime such that Product_{prime p<=p(n)} 1/(1-1/p) >= n. Then lim_{n->oo} p(n+1)/p(n) = exp(exp(-gamma)).
%C A360895 Proof. Follow Mertens's Third Theorem Product_{p<=x} 1/(1-1/p) ~ log(x)/exp(-gamma).
%C A360895 For any particular integer n, it follows from the equations n = log(x_n)/exp(-gamma) -> x_n = exp(n*exp(-gamma)) and n+1 = log(x_n+1)/exp(-gamma) -> x_n+1 = exp((n+1)*exp(-gamma)) that lim_{n->oo} exp((n+1)*exp(-gamma))/exp((n)*exp(-gamma)) = exp(exp(-gamma)).
%C A360895 Convergence table:
%C A360895 n   p(n)        truncated Euler product up to p(n)   ratio p(n)/p(n-1)
%C A360895 42  17427088769 42.0000000010939727723681242652955   1.7532416978341651
%C A360895 43  30553756811 43.0000000012946363551468233325186   1.7532335558736718
%C A360895 44  53567706007 44.0000000002803554088007272169139   1.7532281329055578
%C A360895 45  93916601047 45.0000000002681963271546340553884   1.7532317145469581
%C A360895 46 164657625967 46.0000000002470257389410099668348   1.7532323799133028
%C A360895 47 288682860119 47.0000000001305074313442036255929   1.7532310357544971
%C A360895 48 506127311983 48.0000000000705764045487316221655   1.7532295189758258
%C A360895 oo  oo           oo                                  1.7532294434956945
%H A360895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mertens%27_theorems">Mertens' theorems</a>
%F A360895 Equals exp(A080130).
%F A360895 Limit_{n->oo} A091440(n+1)/A091440(n).
%F A360895 Limit_{n->oo} A061556(n+1)/A061556(n).
%F A360895 Limit_{n->oo} A167348(n+1)/A167348(n).
%e A360895 1.753229443495694582297365429644...
%t A360895 RealDigits[N[Exp[Exp[-EulerGamma]], 115]][[1]]
%o A360895 (PARI) exp(exp(-Euler)) \\ _Michel Marcus_, Feb 25 2023
%Y A360895 Cf. A001620, A080130, A061556, A091440, A167348.
%K A360895 cons,nonn
%O A360895 1,2
%A A360895 _Artur Jasinski_, Feb 25 2023