A244272 - OEIS

%I #11 Sep 08 2022 08:46:08

%S 2,5,6,9,5,4,4,3,4,4,8,9,8,5,7,3,6,0,7,3,7,0,0,4,4,1,1,1,9,3,9,1,4,5,

%T 6,1,6,9,8,8,8,5,8,3,9,6,7,2,6,0,9,5,6,0,3,8,4,2,0,7,7,9,6,4,2,2,5,2,

%U 8,5,3,4,1,2,9,1,8,3,5,2,9,9,8,9,4,9,4,6,9,4,0,4,5,6,9,6,7,4,9,3,6,9,6

%N Decimal expansion of exp(gamma)/log(2), a conjectural constant related to the asymptotic counting of Mersenne primes, where gamma is Euler's constant.

%C Let M(n) denote the number of primes p <= n, for which 2^p-1 is prime. It is conjectured that M(n) -> infinity and that M(n) ~ exp(gamma)/log(2)*log(n).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5 Euler-Mascheroni Constant, p. 29.

%H G. C. Greubel, <a href="/A244272/b244272.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 2.569544344898573607370044111939...

%t RealDigits[Exp[EulerGamma]/Log[2], 10, 103] // First

%o (PARI) default(realprecision, 100); exp(Euler)/log(2) \\ _G. C. Greubel_, Sep 06 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Exp(EulerGamma(R))/Log(2); // _G. C. Greubel_, Sep 06 2018

%Y Cf. A000043, A001620, A073004.

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jun 24 2014