1,1

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.

Eric Weisstein's World of Mathematics, Prime Counting Function.

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

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.

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

Adjacent sequences: A086508 A086509 A086510 this_sequence A086512 A086513 A086514

Sequence in context: A109188 A002532 A098518 this_sequence A138912 A002747 A110377

easy,more,nonn

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