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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.
0
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
0,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.
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.
EXAMPLE
0.3036552633952545488542057678902065632735...
1/xi = 1.4762287836208969657929439948482332947971...
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
CROSSREFS
Cf. A084256 (xi).
Sequence in context: A282610 A011076 A200278 * A010599 A226568 A038517
KEYWORD
nonn,cons
AUTHOR
STATUS
approved