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%I #42 Nov 25 2020 04:58:33
%S 5,1,6,4,5,4,1,7,8,9,4,0,7,8,8,5,6,5,3,3,0,4,8,7,3,4,2,9,7,1,5,2,2,8,
%T 5,8,8,1,5,9,6,8,5,5,3,4,1,5,4,1,9,7,0,1,4,4,1,9,3,1,0,6,5,2,7,3,5,6,
%U 8,7,0,1,4,4,0,2,1,2,7,2,3,4,9,9,1,5,4,8,8,3,2,9,3,6,6,6,2,1,5,3,7,4,0,3,2,4
%N Decimal expansion of sum of the reciprocals of the Mersenne primes.
%C We know this a priori to be strictly less than the Erdős-Borwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.
%H Peter B. Borwein, <a href="https://doi.org/10.1017/S030500410007081X">On the Irrationality of Certain Series</a>, Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1948-04.pdf">On Arithmetical Properties of Lambert Series</a>, J. Indian Math. Soc. 12, 63-66, 1948.
%H Yoshihiro Tanaka, <a href="https://doi.org/10.4236/ajcm.2017.72012">On the Sum of Reciprocals of Mersenne Primes</a>, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erdos-BorweinConstant.html">Erdos-Borwein Constant</a>.
%H Marek Wolf, <a href="http://arxiv.org/abs/1112.2412">Computer experiments with Mersenne primes</a>, arXiv preprint arXiv:1112.2412 [math.NT], 2011.
%F Sum_{i>=1} 1/A000668(i).
%e Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
%e This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).
%p Digits := 120 ; L := [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 ] ;
%p x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i,L)-1 ); end do ;
%t RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* _Amiram Eldar_, May 24 2020 *)
%o (PARI) isM(p)=my(m=Mod(4,2^p-1));for(i=1,p-2,m=m^2-2);!m
%o s=1/3;forprime(p=3,default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p-1)));s \\ _Charles R Greathouse IV_, Mar 22 2012
%Y Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.
%K cons,nonn
%O 0,1
%A _Jonathan Vos Post_, Mar 01 2010
%E Entry revised by _N. J. A. Sloane_, Mar 10 2012
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