

A173898


Decimal expansion of sum of the reciprocals of the Mersenne primes.


6



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, 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, 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
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OFFSET

0,1


COMMENTS

We know this a priori to be strictly less than the ErdősBorwein constant (A065442), which Erdős (1948) showed to be irrational. This new constant would also seem to be irrational.


LINKS

Table of n, a(n) for n=0..105.
Peter B. Borwein, On the Irrationality of Certain Series, Math. Proc. Cambridge Philos. Soc. 112, 141146, 1992.
Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 6366, 1948.
Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145148.
Eric Weisstein's World of Mathematics, ErdosBorwein Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.


FORMULA

Sum_{i>=1} 1/A000668(i).


EXAMPLE

Decimal expansion of (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) + (1/524287) + ... = .5164541789407885653304873429715228588159685534154197.
This has continued fraction expansion 0 + 1/(1 + 1/(1 + 1/(14 + 1/(1 + ...)))) (see A209601).


MAPLE

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 ] ;
x := 0 ; for i from 1 to 30 do x := x+1.0/(2^op(i, L)1 ); end do ;


MATHEMATICA

RealDigits[Sum[1/(2^p  1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)


PROG

(PARI) isM(p)=my(m=Mod(4, 2^p1)); for(i=1, p2, m=m^22); !m
s=1/3; forprime(p=3, default(realprecision)*log(10)\log(2), if(isM(p), s+=1./(2^p1))); s \\ Charles R Greathouse IV, Mar 22 2012


CROSSREFS

Cf. A209601, A000668, A065442 (decimal expansion of ErdosBorwein constant), A000043, A001348, A046051, A057951A057958, A034876, A124477, A135659, A019279, A061652, A000225.
Sequence in context: A086231 A201419 A163336 * A200644 A318265 A318553
Adjacent sequences: A173895 A173896 A173897 * A173899 A173900 A173901


KEYWORD

cons,nonn,changed


AUTHOR

Jonathan Vos Post, Mar 01 2010


EXTENSIONS

Entry revised by N. J. A. Sloane, Mar 10 2012


STATUS

approved



