

A086511


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


1



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
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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 nth 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].
Contribution 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

Cf. A038623, A038624, A038625, A038626, A038627.
Sequence in context: A128239 A307400 A323682 * A291632 A328281 A324372
Adjacent sequences: A086508 A086509 A086510 * A086512 A086513 A086514


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



