OFFSET
1,1
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.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Prime Counting Function.
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)
EXAMPLE
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.
PROG
(PARI) a(n) = { k = 2; while (k <= n*primepi(k), k++); return (k); } \\ Michel Marcus, Jun 19 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Tim Paulden (timmy(AT)cantab.net), Sep 09 2003
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
STATUS
approved