[seqfan] Re: Decimal expansion of sum of the reciprocals of the Mersenne primes

[Charles Greathouse]

I can't speak to the value of having such a sequence, but the constant
can be computed easily:
0.5164541789407885653304873429715228588159685534154197014419310652735687014402127234991548832936662153740324321108365695754191404709248683174860372852946416342768749479588086071312432572644026013151419588344416587074623470840576347668192751532837300659266243983775248186854072542807357246320995557189462157741048445722077812438156511620608464340642405419659689894493689556032048086665764243051362119581186191695180778694367246651921882081430362795649114826030179042078354367049767401292128487279460602677778432373061036262442771921808497109897323519142143128651528968181254217784795527606383671120344719842498930807068367106441291235592492318380299354951485898536549993072642853626168211459888942866276951011775736107716419465282094368753214088532196841537105612237499847833288804729677270380321189748627393750484127221684115025091977819427831975997886489631447813100965300278886360352692663599699364001279233562609577711314882451686385581751514719131451305054209464034126434345506424976281423560789998...

Knowing the Mersenne prime exponents up to p > 2 gives the constant to
at least p bits of precision (since p+1 is not the exponent of a
Mersenne prime and sum(i=p+2, infty, 1/(2^i - 1)) <= sum(i=p+1, infty,
2^-i) = 2^-p).  With A000043 known up to a(39) = 13466917, that's 4
million digits.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Mar 1, 2010 at 12:03 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> I do not see in OEIS either the Decimal expansion nor continued
> fraction representations
> SUM[i=1..infinity] 1/A000668(i) =  SUM[i=1..infinity] reciprocal of
> i-th Mersenne prime (primes of form 2^p - 1 where p is a prime)
>
> (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) +  (1/524287) + ...
>
> We know it to be strictly less than the Erdős-Borwein constant,which,
> is the sum of the reciprocals of the Mersenne numbers.
>
> The crudest calculation (using Google as a low-precision calculator) gives
> (1/3) + (1/7) + (1/31) + (1/127) + (1/8191) + (1/131071) ~ 0.516452271
>
> Have I missed it as a seq, or is it worth doing to 100 digits as a new seq?
>
>
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