cp's OEIS Frontend

A190303 Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).

%I A190303 #26 Feb 16 2025 08:33:14
%S A190303 4,4,6,6,8,4,3,0,7
%N A190303 Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).
%C A190303 Computed 0.446684 for n = 1 to 65536, using Open Office Calc. Next digit expected to be between 2 and 3.
%C A190303 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.
%C A190303 1          1      0.50000000000000000
%C A190303 2          9     -0.05765566386047510
%C A190303 3         62      0.00388002010130731
%C A190303 4        487      0.00050881775862179
%C A190303 5       3900     -0.00004384563815649
%C A190303 6      32501     -0.00000552572415587
%C A190303 7     279106      0.00000045427780897
%C A190303 8    2444255      0.00000005495474474
%C A190303 9   21731345     -0.00000000549864067
%C A190303 Total:            0.44668430669928564 - _T. D. Noe_, May 08 2011
%C A190303 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]
%H A190303 J. Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010.
%H A190303 J. Sondow, <a href="http://www.jstor.org/stable/40391170">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly 116 (2009), 630-635.
%H A190303 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2
%H A190303 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%F A190303 Sum_{n>=1} (-1)^(n-1)(1/R_n), where R_n is the n-th Ramanujan prime, A104272(n).
%e A190303 0.446684307...
%Y A190303 Cf. A104272, A085548, A078437, A190124.
%K A190303 nonn,cons,more
%O A190303 0,1
%A A190303 _John W. Nicholson_, May 07 2011
%E A190303 Definition corrected by _Jonathan Sondow_, May 10 2011