A290394 - OEIS

%I #20 Jan 01 2018 13:17:41

%S 2,11,11,29,29,37,37,53,127,127,127,127,127,149,149,149,211,223,223,

%T 223,307,307,331,331,331,331,331,331,331,541,541,541,541,541,541,541,

%U 541,541,541,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1361,1693

%N First k-Ramanujan prime, where k = 1 + 1/n.

%C For real k > 1, the first k-Ramanujan prime is the smallest integer m with pi(x) - pi(x/k) >= 1 for all real x >= m. For 0 < c < 1, the first c-Ramanujan prime is the first k-Ramanujan prime with k = 1/c.

%C Axler (2015, Cor. 2.4 and Prop. 2.5(ii)) and Axler and Leßmann (2017, Theorem 1) computed the first k-Ramanujan prime for all k >= 1.000040690557321. With k = 1 + 1/n, this gives 1 <= n <= 24575; in particular, a(24575) = 2898359. They also give the isolated result a(28313999) = 10726905041 on p. 646.

%C The Mathematica program below is based on their algorithm but uses only part of their data (compare A277719) and is valid only for 1 <= n <= 1014; in particular, a(1014) = 48731. Their algorithm uses their result that for N > 1 the N-th prime p_N is the first k-Ramanujan prime if and only if p_N > k*p_{N-1} and p_n <= k*p_{n-1} for all n > N.

%C See A104272 for additional comments, references, links, formulas, examples, programs, and cross-refs.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13; <a href="http://arxiv.org/abs/1108.0475">arXiv:1108.0475 [math.NT], 2011</a>.

%H Christian Axler, <a href="http://dx.doi.org/10.1007/s11139-015-9693-9">On generalized Ramanujan primes</a>, Ramanujan J., online 30 April 2015, 1-30.

%H Christian Axler and Thomas Leßmann, <a href="http://arxiv.org/abs/1504.05485">An explicit upper bound for the first k-Ramanujan prime</a>, arXiv:1504.05485 [math.NT], 2015.

%H Christian Axler and Thomas Leßmann, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.7.642">On the first k-Ramanujan prime</a>, Amer. Math. Monthly, 124 (2017), 642-646; correction by J. Sondow, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.10.983">Editor's endnotes</a>, Amer. Math. Monthly, 124 (2017), 985.

%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4.

%e a(1) = first 2-Ramanujan prime = first 1/2-Ramanujan prime = first Ramanujan prime = A104272(1) = 2.

%e a(3) = first 4/3-Ramanujan prime = first 3/4-Ramanujan prime = A193880(1) = 11.

%t A = {3, 5, 7, 10, 12, 16, 31, 35, 47, 48, 63, 67, 100, 218, 264, 298, 328,  368, 430, 463, 591, 651, 739, 758, 782, 843, 891, 929, 1060, 1184, 1230, 1316, 1410, 1832, 2226, 3386, 3645, 3794, 3796, 4523, 4613, 4755, 5009, 5950}; kR1[k_] := If[k >= 5/3, 2, (m = 1;

%t    While[k >= Prime[A[[m]]]/Prime[A[[m]] - 1] ||

%t      k < Prime[A[[m + 1]]]/Prime[A[[m + 1]] - 1], m++];

%t    Prime[A[[m]]])]; Table[kR1[1 + 1/n], {n, 70}]

%Y Cf. A104272, A164952, A193761, A193880, A277718, A277719.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Jul 29 2017