A243350 Decimal expansion of the unique solution of the equation sum_(p prime)(1/p^x) = 1, a constant related to the asymptotic evaluation of the number of prime multiplicative compositions.
1, 3, 9, 9, 4, 3, 3, 3, 2, 8, 7, 2, 6, 3, 3, 0, 3, 1, 8, 2, 0, 2, 8, 0, 7, 2, 1, 4, 7, 4, 5, 6, 4, 4, 3, 2, 7, 9, 0, 4, 7, 2, 7, 4, 2, 9, 4, 8, 4, 3, 8, 3, 9, 4, 1, 2, 7, 4, 7, 6, 5, 8, 2, 2, 8, 8, 8, 0, 6, 2, 4, 9, 2, 4, 8, 7, 2, 4, 7, 8, 0, 0, 2, 3, 3, 3, 9, 0, 5, 2, 0, 0, 2, 1, 6, 6, 8, 5, 1, 3
Offset: 1
Examples
1.3994333287263303182028072147456443279...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's composition constant, p. 293.
Links
- Jean-Francois Alcover, Table of n, a(n) for n = 1..100
- Hugh L. Montgomery and Gérald Tenenbaum, On multiplicative compositions of integers, Mathematika 63:3 (2017), pp. 1081-1090.
- Eric Weisstein's MathWorld, Prime Zeta function
Crossrefs
Cf. A243584.
Programs
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Mathematica
digits = 100; eta = x /. FindRoot[PrimeZetaP[x] == 1, {x, 3/2}, WorkingPrecision -> digits+5]; RealDigits[eta, 10, digits] // First -
PARI
eps(x=1.)=my(p=if(x,precision(x),default(realprecision)));precision(2.>>(32*ceil(p*38539962/371253907))*abs(x),9) primezeta(s)=my(t=s*log(2),iter=lambertw(t/eps())\t,tot); forsquarefree(k=1,iter, tot+=moebius(k)/k[1]*log(abs(zeta(k[1]*s)))); tot; solve(x=1.399,1.4,primezeta(x)-1) \\ Charles R Greathouse IV, Nov 16 2018
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PARI
solve(x=1.05,1.5,1-sumeulerrat(1/p,x)) \\ Hugo Pfoertner, Nov 28 2021