

A290394


First kRamanujan prime, where k = 1 + 1/n.


5



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

For real k > 1, the first kRamanujan prime is the smallest integer m with pi(x)  pi(x/k) >= 1 for all real x >= m. For 0 < c < 1, the first cRamanujan prime is the first kRamanujan 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 kRamanujan 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 Nth prime p_N is the first kRamanujan prime if and only if p_N > k*p_{N1} and p_n <= k*p_{n1} for all n > N.
See A104272 for additional comments, references, links, formulas, examples, programs, and crossrefs.


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), 113; arXiv:1108.0475 [math.NT], 2011.
Christian Axler, On generalized Ramanujan primes, Ramanujan J., online 30 April 2015, 130.
Christian Axler and Thomas Leßmann, An explicit upper bound for the first kRamanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first kRamanujan prime, Amer. Math. Monthly, 124 (2017), 642646; 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 2Ramanujan prime = first 1/2Ramanujan prime = first Ramanujan prime = A104272(1) = 2.
a(3) = first 4/3Ramanujan prime = first 3/4Ramanujan 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



