A220293 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.

11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 223, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 443, 461, 487, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653

Offset

1,1

Comments

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)).

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.

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

Links

Table of n, a(n) for n=1..50.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv 2011.

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

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Formula

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

Crossrefs

Cf. A104272.

Sequence in context: A225493 A051634 A038918 * A166307 A128464 A105170

Adjacent sequences: A220290 A220291 A220292 * A220294 A220295 A220296

Keyword

nonn

Author

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 09 2012

Status

approved