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Decimal expansion of the coefficient c appearing in the asymptotic evaluation of the number of prime additive compositions of n as c*(1/xi)^n, where xi is A084256.
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%I #5 Sep 23 2014 11:19:29

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

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

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

%N Decimal expansion of the coefficient c appearing in the asymptotic evaluation of the number of prime additive compositions of n as c*(1/xi)^n, where xi is A084256.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

%F c = 1/(xi*f'(xi)), where f(x) is the sum over primes x^2 + x^3 + x^5 + x^7 + ..., xi (A084256) being the positive solution of f(x) = 1.

%e 0.3036552633952545488542057678902065632735...

%e 1/xi = 1.4762287836208969657929439948482332947971...

%t nMax = 200; digits = 102; f[x_] := Sum[x^Prime[n], {n, 1, nMax}]; fp[x_] := Sum[Prime[n]*x^(Prime[n] - 1), {n, 1, nMax}]; xi = x /. FindRoot[f[x] == 1, {x, 2/3}, WorkingPrecision -> digits+5]; c = 1/(xi*fp[xi]); RealDigits[c, 10, digits] // First

%Y Cf. A084256 (xi).

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Sep 23 2014