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A173898
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Decimal expansion of sum of the reciprocals of the Mersenne primes.
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11
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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
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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).
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MAPLE
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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 ;
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MATHEMATICA
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RealDigits[Sum[1/(2^p - 1), {p, MersennePrimeExponent[Range[14]]}], 10, 100][[1]] (* Amiram Eldar, May 24 2020 *)
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PROG
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(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
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CROSSREFS
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Cf. A209601, A000668, A065442 (decimal expansion of Erdos-Borwein constant), A000043, A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000225.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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