A220293 - OEIS

%I #46 Nov 08 2014 18:52:52

%S 11,17,29,41,47,59,67,71,97,101,107,127,149,151,167,179,223,229,233,

%T 239,241,263,269,281,307,311,347,349,367,373,401,409,419,431,433,443,

%U 461,487,503,569,571,587,593,599,601,607,641,643,647,653

%N Chebyshev numbers C_2(n): a(n) is the smallest number such that if x >= a(n), then theta(x) - theta(x/2) >= n*log(x), where theta(x) = sum_{prime p <= x} log p.

%C Up to a(100)=1489, only two terms of the sequence (a(17)=223 and a(36)=443) are not Ramanujan numbers (A104272), and the sequence is missing only the following Ramanujan numbers up to 1489: 2, 181, 227, 439, 491, 1283, and 1301. The latter observation shows how closely the ratio theta(x)/log(x) approximates the number of primes <= x (i.e., pi(x)).

%C A generalization: for a real number v>1, the v-Chebyshev number C_v(n) is the smallest integer k such that if x>=k, then theta(x)-theta(x/v)>=n*log x. In particular, a(n)=C_2(n). For another example, if v=4/3, then, at least up to 3319, all (4/3)-Chebyshev numbers are (4/3)-Ramanujan primes as in Shevelev's link (cf. A193880, where c=1/v=3/4 is excepted), and in this case the sequence is missing only the following (4/3)-Ramanujan numbers up to 3319: 11 and 1567.

%C Like Chebyshev numbers, all v-Chebyshev numbers are primes.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv 2011.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://link.springer.com/chapter/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

%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

%H Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>

%F For n >= 2, A104272(n) <= a(n-1) <= prime(3n).

%Y Cf. A104272.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 09 2012