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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.
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%I #39 Nov 29 2021 01:33:00

%S 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,

%T 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,

%U 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

%N 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.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's composition constant, p. 293.

%H Jean-Francois Alcover, <a href="/A243350/b243350.txt">Table of n, a(n) for n = 1..100</a>

%H Hugh L. Montgomery and Gérald Tenenbaum, <a href="http://www.iecl.univ-lorraine.fr/~Gerald.Tenenbaum/PUBLIC/PPP/Thompson.pdf">On multiplicative compositions of integers</a>, Mathematika 63:3 (2017), pp. 1081-1090.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta function</a>

%e 1.3994333287263303182028072147456443279...

%t digits = 100; eta = x /. FindRoot[PrimeZetaP[x] == 1, {x, 3/2}, WorkingPrecision -> digits+5]; RealDigits[eta, 10, digits] // First

%o (PARI) eps(x=1.)=my(p=if(x,precision(x),default(realprecision)));precision(2.>>(32*ceil(p*38539962/371253907))*abs(x),9)

%o 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;

%o solve(x=1.399,1.4,primezeta(x)-1) \\ _Charles R Greathouse IV_, Nov 16 2018

%o (PARI) solve(x=1.05,1.5,1-sumeulerrat(1/p,x)) \\ _Hugo Pfoertner_, Nov 28 2021

%Y Cf. A243584.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Jun 06 2014