

A190303


Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(1)^(n1), where R_n is the nth Ramanujan prime, A104272(n).


0




OFFSET

0,1


COMMENTS

Computed 0.446684 for n = 1 to 65536, using Open Office Calc. Next digit expected to be between 2 and 3.
By computing all Ramanujan primes less than 10^9, we find that about 9 decimal places of the sum should be correct: 0.446684307 (truncated, not rounded). The following table shows the number of Ramanujan primes between powers of 10 and the sum of the alternating reciprocals of those primes.
1 1 0.50000000000000000
2 9 0.05765566386047510
3 62 0.00388002010130731
4 487 0.00050881775862179
5 3900 0.00004384563815649
6 32501 0.00000552572415587
7 279106 0.00000045427780897
8 2444255 0.00000005495474474
9 21731345 0.00000000549864067
Total: 0.44668430669928564  T. D. Noe, May 08 2011
Let E_n denote the error after the first n terms in the series. Then by the Alternating Series Test, 1/R_{n+1}  1/R_{n+2} < E_n < 1/R_{n+1}. [Jonathan Sondow, May 10 2011]


LINKS

Table of n, a(n) for n=0..8.
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 20092010.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
Eric W. Weisstein, Prime Sums


FORMULA

Sum_{n>=1} (1)^(n1)(1/R_n), where R_n is the nth Ramanujan prime, A104272(n).


EXAMPLE

0.446684307...


CROSSREFS

Cf. A104272, A085548, A078437, A190124.
Sequence in context: A111657 A049751 A196594 * A175123 A276679 A163638
Adjacent sequences: A190300 A190301 A190302 * A190304 A190305 A190306


KEYWORD

nonn,cons,more


AUTHOR

John W. Nicholson, May 07 2011


EXTENSIONS

Definition corrected by Jonathan Sondow, May 10 2011


STATUS

approved



