A247734 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.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Sep 23 2014

Keywords

• nonn
• cons

Examples

			0.3036552633952545488542057678902065632735...
1/xi = 1.4762287836208969657929439948482332947971...
		

References

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

Crossrefs

Cf. A084256 (xi).

Programs

Mathematica

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

Formula

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.