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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. 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 (list; graph; refs; listen; history; text; internal format)
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].

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

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Last modified November 22 03:43 EST 2019. Contains 329388 sequences. (Running on oeis4.)