A182072 a(n) is the minimal m such that sqrt(m) - pi(sqrt(p_m)) >= n, where p_m is the m-th prime.

1, 121, 144, 256, 400, 484, 784, 961, 1089, 1156, 1681, 1764, 1849, 2116, 2401, 2704, 3025, 3364, 3721, 3844, 4489, 4624, 4900, 5041, 5776, 6241, 6561, 7056, 8100, 8281, 8649, 8836, 9025, 10000, 10201, 10404, 11025, 11236, 12321, 12544, 13225, 13924, 14400

Offset

1,2

Comments

All terms are squares.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

a(n)~n^2 as n tends to infinity. Indeed, by the PNT, we have pi(sqrt(p_m)) ~ 2*sqrt(p_m)/log(p_m) ~ 2*sqrt(m*log(m))/log(m)=2*sqrt(m/log(m)). Thus, if sqrt(m)-2*sqrt(m/log(m)) = sqrt(m)*(1-2/sqrt(log(m))) = n, then m ~ n^2.

Example

a(2)=121, since p_121 = 661 and sqrt(121)-pi(sqrt(661)) = 11- pi(25) = 11 - 9 = 2, while p_120 = 659 and sqrt(120)-pi(sqrt(659)) = sqrt(120)-9 < 2.

Mathematica

Module[{last = 1}, Table[last = NestWhile[#1 + 1 &, last, Sqrt[#1] - PrimePi[Floor[Sqrt[Prime[#1]]]] < n &], {n, 1, 55}]]

Crossrefs

Cf. A000040, A000290.

Sequence in context: A287390 A258202 A242372 * A262514 A258693 A037266

Adjacent sequences: A182069 A182070 A182071 * A182073 A182074 A182075

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Apr 10 2012

Status

approved