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A240953 Constant in Sathe's theorem: Product_{p prime} (1 - 1/p)*e^(1/p).

%I A240953 #32 Nov 18 2018 12:24:55
%S A240953 7,2,9,2,6,4,7,4,4,2,5,7,1,1,9,0,1,8,8,5,3,6,1,5,3,1,6,9,3,1,3,0,0,1,
%T A240953 2,8,1,7,7,5,4,5,9,7,1,0,3,7,8,4,3,6,1,8,6,7,4,7,6,6,9,1,2,8,7,6,5,5,
%U A240953 6,4,6,6,1,2,5,6,6,7,2,2,9,4,7,4,2,8,3,5,9,1,5,6,4,2,8,0,1,6,9,7,4,7,2
%N A240953 Constant in Sathe's theorem: Product_{p prime} (1 - 1/p)*e^(1/p).
%C A240953 Sathe proved that pi_k(x), the count of numbers <= x with exactly k prime factors, satisfies pi_k(x) ~ f(k/log log x) * x/log x * (log log x)^(k-1)/(k-1)! where f(x) = c/gamma(x+1) * Product_{p prime} 1 + x*exp(-x/p)/p and c is this constant.  This holds uniformly for k < (2 - eps)log log x for any fixed eps > 0. - _Charles R Greathouse IV_, Aug 02 2016
%D A240953 L. G. Sathe, On a problem of Hardy on the distribution of integers having a given number of prime factors. I., J. Indian Math. Soc. (N.S.) 17 (1953), pp. 63-82.
%D A240953 L. G. Sathe, On a problem of Hardy on the distribution of integers having a given number of prime factors. II., J. Indian Math. Soc. (N.S.) 17 (1953), pp. 83-141.
%D A240953 Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. (N.S.) 18 (1954), pp. 83-87.
%F A240953 Equals e^A143524. - _Jon Maiga_, Nov 17 2018
%e A240953 0.72926474425711901885361531693130012817754597103784361867476691287655...
%t A240953 digits = 103; S = E^-NSum[PrimeZetaP[ n]/n, {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[S, 10, digits] // First (* _Jean-François Alcover_, Sep 11 2015 *)
%o A240953 (PARI) /* Helper functions and a function f to compute a k-th order approximation of the constant using the primes up to lim. */
%o A240953 eps(x=1.)=my(p=if(x,precision(x),default(realprecision))); precision(2. >> (32 * ceil(p * 38539962 / 371253907)), 9);
%o A240953 primezeta(s)=my(lm=s*log(2));lm=lambertw(lm/eps())\lm;sum(k=1,lm,moebius(k)/k*log(abs(zeta(k*s))));
%o A240953 f(lim,k)=my(t=0.);forprime(p=2,lim,t+=log(1-1/p)+sum(i=1,k,1/i/p^i));exp(t-sum(i=2,k,primezeta(i)/i));
%o A240953 f(1e8, 9)
%Y A240953 Cf. A001222, A143524.
%K A240953 nonn,cons
%O A240953 0,1
%A A240953 _Charles R Greathouse IV_, Aug 04 2014
%E A240953 More digits from _Jean-François Alcover_, Sep 11 2015