

A225907


Smallest nRamanujan prime that is less than half of the next nRamanujan prime, or 0 if none exists.


4




OFFSET

0,2


COMMENTS

In A192824 Noe defines 0Ramanujan primes to be simply primes, and 1Ramanujan primes to be Ramanujan primes. Define the kth 2Ramanujan prime to be the smallest number R'_k (the notation in Paksoy 2012) with the property that the interval (x/2,x] contains at least k 1Ramanujan primes, for any x >= R'_k. Continuing inductively, define nRamanujan primes in terms of (n1)Ramanujan primes.
Only the first three terms 0, 2, 11 are proved (by Chebyshev, Ramanujan, and Paksoy, respectively). The rest are conjecturalsee the 2nd comment in A192821.
See A104272 for additional comments, references, links, and crossrefs.
Is it true that for every n there exists K = K(n) such that for all k > K, the kth nRamanujan prime is greater than half of the (k+1)th nRamanujan prime? (Equivalently, is there a largest nRamanujan prime that is less than half of the next nRamanujan prime?) It is true for n = 0 by Bertrand's Postulate (see A062234), and for n = 1 by a theorem of Paksoy. Is it even true that if n is fixed, then (kth nRamanujan prime) ~ ((k+1)th nRamanujan prime) as k > infinity?  Jonathan Sondow, Dec 16 2013


LINKS

Table of n, a(n) for n=0..5.
Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv 2012.


EXAMPLE

By Bertrand's Postulate (proved by Chebyshev), prime(k+1) < 2*prime(k) for all k, so a(0) = 0.
Ramanujan proved that the Ramanujan primes begin 2, 11, ..., so a(1) = 2.
Paksoy proved that the 2Ramanujan primes begin 11, 41,..., so a(2) = 11.
It appears that the 3Ramanujan primes begin 41, 149, ...; if true, then a(3) = 41.
It appears that the 4Ramanujan primes begin 569, 571, 587, 1367 ...; if true, then a(4) = 587.


CROSSREFS

Cf. A000040 (0Ramanujan primes), A104272 (1Ramanujan primes), A192820 (2Ramanujan primes), A192821 (3Ramanujan primes), A192822 (4Ramanujan primes), A192823 (5Ramanujan primes), A192824 (least nRamanujan prime). Cf. also A233822 = 2*R(n)  R(n+1) and A062234.
Sequence in context: A260267 A128241 A258937 * A107020 A160945 A079808
Adjacent sequences: A225904 A225905 A225906 * A225908 A225909 A225910


KEYWORD

nonn,more


AUTHOR

Jonathan Sondow, Jun 08 2013


STATUS

approved



