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A173898 Decimal expansion of sum of the reciprocals of the Mersenne primes. 5
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 Erdos-Borwein constant (A065442), which Erdos (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 Erdos, On Arithmetical Properties of Lambert Series, J. Indian Math. Soc. 12, 63-66, 1948.

Eric W. Weisstein, 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 ;

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

AUTHOR

Jonathan Vos Post, Mar 01 2010

EXTENSIONS

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

STATUS

approved

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Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)