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

%I A086511 #22 Feb 16 2025 08:32:50
%S A086511 2,9,28,121,336,1081,3060,8409,23527,64541,175198,480865,1304499,
%T A086511 3523885,9557956,25874753,70115413,189961183,514272412,1394193581,
%U A086511 3779849620,10246935645,27788566030,75370121161,204475052376,554805820453,1505578023622,4086199301997
%N A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.
%C A086511 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.
%H A086511 Giovanni Resta, <a href="/A086511/b086511.txt">Table of n, a(n) for n = 1..50</a>
%H A086511 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>.
%F A086511 Heuristically, for large n, a(n) ~= 3.0787*(2.70888^n) [error < 0.05% for 15 <= n <= 20].
%F A086511 From _Nathaniel Johnston_, Apr 10 2011: (Start)
%F A086511 a(n) >= exp(n/2 + sqrt(n^2 + 4n)/2), n >= 6.
%F A086511 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, ...
%F A086511 (End)
%e A086511 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.
%o A086511 (PARI) a(n) = { k = 2; while (k <= n*primepi(k), k++); return (k);} \\ _Michel Marcus_, Jun 19 2013
%Y A086511 Cf. A038623, A038624, A038625, A038626, A038627.
%K A086511 nonn
%O A086511 1,1
%A A086511 Tim Paulden (timmy(AT)cantab.net), Sep 09 2003
%E A086511 a(21)-a(26) from _Nathaniel Johnston_, Apr 10 2011
%E A086511 Corrected a(26) and a(27)-a(28) from _Giovanni Resta_, Sep 01 2018
%E A086511 a(29)-a(50) obtained from the values of A038625 computed by _Jan Büthe_. - _Giovanni Resta_, Sep 01 2018