A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

2, 9, 28, 121, 336, 1081, 3060, 8409, 23527, 64541, 175198, 480865, 1304499, 3523885, 9557956, 25874753, 70115413, 189961183, 514272412, 1394193581, 3779849620, 10246935645, 27788566030, 75370121161, 204475052376, 554805820453, 1505578023622, 4086199301997

Offset: 1

Author

Tim Paulden (timmy(AT)cantab.net), Sep 09 2003

Keywords

• nonn

Comments

a(n) is bounded above by the sequence A038623, in which k is required to be prime. In addition, the sequence pi(a(n)) = {1, 4, 9, 30, 67, 180, 437, 1051, ...} closely resembles the sequence A038624, in which the n-th term is the minimal t such that k >= n * pi(k) for every k satisfying pi(k) = t. If we were to make the inequality in A038624 strict, the resulting sequence would provide an upper bound for pi(a(n)). Sequences A038625, A038626 and A038627 focus on the equality k = n * pi(k): as we would expect, a(n) follows A038625 very closely for large n.

Examples

			Consider the pairs (k, pi(k)) for k > 1. The inequality k > 1 * pi(k) is first satisfied at k = 2 and so a(1) = 2. Similarly, the inequality k > 2 * pi(k) is first satisfied at k = 9 and so a(2) = 9.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Prime Counting Function.

Crossrefs

Cf. A038623, A038624, A038625, A038626, A038627.

Programs

PARI

a(n) = { k = 2; while (k <= n*primepi(k), k++); return (k);} \\ Michel Marcus, Jun 19 2013

Formula

Heuristically, for large n, a(n) ~= 3.0787*(2.70888^n) [error < 0.05% for 15 <= n <= 20].

From Nathaniel Johnston, Apr 10 2011: (Start)

a(n) >= exp(n/2 + sqrt(n^2 + 4n)/2), n >= 6.

a(n) = A038625(n) + m(n)*n + 1 for some m(n) >= 0. For n = 2, 3, 4, ..., m(n) = 3, 0, 6, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...

(End)

Extensions

a(21)-a(26) from Nathaniel Johnston, Apr 10 2011

Corrected a(26) and a(27)-a(28) from Giovanni Resta, Sep 01 2018

a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018