login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

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, 141-146, 1992.

Paul Erdős, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.

Yoshihiro Tanaka, On the Sum of Reciprocals of Mersenne Primes, American Journal of Computational Mathematics, Vol. 7, No. 2 (2017), pp. 145-148.

Eric Weisstein's World of Mathematics, Erdos-Borwein 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^p-1)); for(i=1, p-2, m=m^2-2); !m

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

CROSSREFS

Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 15:14 EST 2020. Contains 338807 sequences. (Running on oeis4.)