A290394 First k-Ramanujan prime, where k = 1 + 1/n.

2, 11, 11, 29, 29, 37, 37, 53, 127, 127, 127, 127, 127, 149, 149, 149, 211, 223, 223, 223, 307, 307, 331, 331, 331, 331, 331, 331, 331, 541, 541, 541, 541, 541, 541, 541, 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

Offset: 1

Author

Jonathan Sondow, Jul 29 2017

Keywords

• nonn

Comments

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.

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.

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.

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

Examples

			a(1) = first 2-Ramanujan prime = first 1/2-Ramanujan prime = first Ramanujan prime = A104272(1) = 2.
a(3) = first 4/3-Ramanujan prime = first 3/4-Ramanujan prime = A193880(1) = 11.
		

Links

• N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13; arXiv:1108.0475 [math.NT], 2011.
• Christian Axler, On generalized Ramanujan primes, Ramanujan J., online 30 April 2015, 1-30.
• Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
• Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646; correction by J. Sondow, Editor's endnotes, Amer. Math. Monthly, 124 (2017), 985.
• V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.

Crossrefs

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

Programs

Mathematica

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;
   While[k >= Prime[A[[m]]]/Prime[A[[m]] - 1] ||
     k < Prime[A[[m + 1]]]/Prime[A[[m + 1]] - 1], m++];
   Prime[A[[m]]])]; Table[kR1[1 + 1/n], {n, 70}]