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

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, 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, 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

Offset: 0

Author

Charles R Greathouse IV, Aug 04 2014

Keywords

• nonn
• cons

Comments

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

Examples

			0.72926474425711901885361531693130012817754597103784361867476691287655...
		

References

• 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.
• 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.
• Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. (N.S.) 18 (1954), pp. 83-87.

Crossrefs

Cf. A001222, A143524.

Programs

Mathematica

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 *)

PARI

/* Helper functions and a function f to compute a k-th order approximation of the constant using the primes up to lim. */
eps(x=1.)=my(p=if(x,precision(x),default(realprecision))); precision(2. >> (32 * ceil(p * 38539962 / 371253907)), 9);
primezeta(s)=my(lm=s*log(2));lm=lambertw(lm/eps())\lm;sum(k=1,lm,moebius(k)/k*log(abs(zeta(k*s))));
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));
f(1e8, 9)

Formula

Equals e^A143524. - Jon Maiga, Nov 17 2018

Extensions

More digits from Jean-François Alcover, Sep 11 2015