A192820 2-Ramanujan primes: the interval (x/2,x] has at least n Ramanujan primes for x >= a(n) but not for x = a(n) - 1.

11, 41, 59, 97, 149, 151, 227, 229, 233, 239, 263, 307, 367, 373, 401, 409, 569, 571, 587, 593, 599, 641, 643, 647, 653, 719, 751, 821, 937, 941, 1009, 1019, 1021, 1031, 1049, 1051, 1061, 1063, 1217, 1367, 1373, 1423, 1427, 1439, 1481, 1487, 1549, 1553, 1559

Offset: 1

Author

T. D. Noe, Jul 11 2011

Keywords

• nonn

Comments

It is conjectured that primepi(a(n)) <= 7*n for all n. - T. D. Noe, Aug 26 2011

Subsequence of the Ramanujan primes A104272, by the minimality of a(n). - Jonathan Sondow, Aug 21 2012

Paksoy (2012) denotes a(n) by R'n and calls it "the n-th derived Ramanujan prime." He proves the bounds on R'_n below. - _Jonathan Sondow, Oct 29 2012

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• M. B. Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

Crossrefs

Cf. A104272 (Ramanujan primes), A192821, A192822, A192823, A192824, A225907.

Formula

R(2n) <= a(n) < R(3n), where R(n) = the n-th Ramanujan prime (Paksoy 2012).

p(4n) < a(n) < p(9n), where p(n) = the n-th prime (Paksoy 2012).

a(n) < p(8n) for n >= 5315 (Paksoy 2012).

R(2n) ~ a(n) ~ p(4n) as n -> oo (Paksoy 2012).

Extensions

Definition clarified by Jonathan Sondow, Aug 21 2012