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

Author

Jean-François Alcover, Jun 06 2014

Keywords

• nonn
• cons

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

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

PARI

solve(x=1.05,1.5,1-sumeulerrat(1/p,x)) \\ Hugo Pfoertner, Nov 28 2021