A162996 a(n) = Round(kn * (log(kn)+1)), with k = 2.216 as an approximation of R_n = n-th Ramanujan prime A104272(n) and Abs(a(n)-R_n) < 2 * Sqrt(a(n)) for n in [1..1000].

4, 11, 19, 28, 38, 48, 58, 69, 80, 91, 102, 114, 126, 138, 150, 162, 174, 187, 200, 212, 225, 238, 251, 265, 278, 291, 305, 318, 332, 345, 359, 373, 387, 401, 415, 429, 443, 458, 472, 486, 501, 515, 530, 544, 559, 573, 588, 603, 618, 632, 647, 662, 677, 692

Offset: 1

Author

Daniel Forgues, Jul 21 2009, Jul 29 2009

Keywords

• nonn

Comments

a(n) approximates the {kn}-th prime number, which in turn approximates the n-th Ramanujan prime, and k = 2.216 is nearly optimal for n in [1..1000] since a(n) - 2*sqrt(a(n)) < R_n < a(n) + 2*sqrt(a(n)) in that range. Since R_n ~ Prime(2n) ~ 2n * (log(2n)+1) ~ 2n * log(2n), whereas A162996(n) ~ Prime(kn) ~ kn * (log(kn)+1) ~ kn * log(kn), giving A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2).

R_n is the smallest number such that if x >= R_n, then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.

Links

• Daniel Forgues, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A163160 (Round(kn * (log(kn)+1)) - R_n, where k = 2.216 and R_n = n-th Ramanujan prime).

Cf. A104272 (Ramanujan primes).