%I #5 Sep 24 2014 05:54:44
%S 6,2,2,3,0,6,5,7,4,5,7,0,0,8,5,6,6,4,6,2,1,3,4,1,1,8,1,2,7,0,0,0,9,6,
%T 0,5,1,3,0,7,8,4,3,0,1,4,7,9,0,0,7,8,5,4,2,0,3,7,4,7,2,8,1,5,6,2,4,6,
%U 0,4,6,7,8,6,9,4,6,2,4,0,8,4,8,9,4,6,3,5,8,8,2,2,0,8,7,6,3,6,8,2
%N Decimal expansion of 1/(theta*P'(theta)), a constant appearing in the asymptotic evaluation of the coefficients q_n in 1/(1+P(x)), where P(x) is the generating function of the primes and theta the unique zero of P(x) in [-3/4, 0].
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 294 and p. 551.
%F q_n ~ (1/(theta*P'(theta))) * (1/theta^n).
%e -0.622306574570085664621341181270009605130784301479...
%t digits = 100; P[x_] := 1 + Sum[Prime[n]*x^n, {n, 1, 1000}]; PPrime[x_] := Sum[n*Prime[n]*x^(n-1), {n, 1, 1000}]; theta = x /. FindRoot[P[x] == 0, {x, -3/4}, WorkingPrecision -> digits+5]; RealDigits[1/(theta*PPrime[theta]), 10, digits] // First
%Y Cf. A030018, A072508, A088751.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Sep 24 2014