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

1,1

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.

Links

Table of n, a(n) for n=1..70.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A265561 A265545 A153705 * A292779 A245521 A275617

Adjacent sequences: A290391 A290392 A290393 * A290395 A290396 A290397

Keyword

nonn

Author

Jonathan Sondow, Jul 29 2017

Status

approved