p[n_, a_] := Exp[-a] Sum[a^Prime[k]/Prime[k]! , {k, 1, n}]

With[{a = 50},
 Table[1 - p[k + 1, a] p[k,a// N, {k, 20, 30}]]

(* {-0.00205371, -0.00016001, -0.0000226721, -8.46669*10^-7, -5.67576*10^-9,
-3.62011*10^-10, -8.61438*10^-11, -4.34702*10^-12, -9.2317*10^-13,
-3.73379*10^-14, -1.68778*10^-19} *)

p(a>>1) ~= 1/(2 Sqrt(a))( (a+Sqrt(a))/Log(a+Sqrt(a)) -(a-Sqrt(a))/Log(a-Sqrt(a)) )

{h[10], h[50], h[100], h[400]} = {0.245097, 0.190413, 0.170051, 0.139057}

p(a->oo) = 1/Log(a)