A038625 a(n) is smallest number m such that m = n*pi(m), where pi(k) = number of primes <= k (A000720).

2, 27, 96, 330, 1008, 3059, 8408, 23526, 64540, 175197, 480852, 1304498, 3523884, 9557955, 25874752, 70115412, 189961182, 514272411, 1394193580, 3779849598, 10246935644, 27788566029, 75370121160, 204475052375, 554805820452, 1505578023621, 4086199301996, 11091501630949

Offset: 2

Author

Jud McCranie

Keywords

• nonn

Comments

Golomb shows that solutions exist for each n>1.

Equivalently, for n > 1, least m such that m >= n*pi(m). - Eric M. Schmidt, Aug 05 2014

The values a(26),...,a(50) were calculated with the Eratosthenes sieve making use of strong bounds for pi(x), which follow from partial knowledge of the Riemann hypothesis, and the analytic method for calculating initial values of pi(x). - Jan Büthe, Jan 16 2015

Examples

			pi(3059) = 437 and 3059/437 = 7, so a(7)=3059.
		

Links

• Jan Büthe, Table of n, a(n) for n = 2..50
• John D. Cook, Integer odds and prime numbers
• S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
• Eric Weisstein's World of Mathematics, Prime Counting Function.

Crossrefs

Cf. A000720, A073436, A038623, A038624, A038625, A038626, A038627, A057809.

Programs

Maple

with(numtheory); f:=proc(n) local i; for i from 2 to 10000 do if i mod pi(i) = 0 and i/pi(i) = n then RETURN(i); fi; od: RETURN(-1); end; # N. J. A. Sloane, Sep 01 2008

Mathematica

t = {}; k = 2; Do[While[n*PrimePi[k] != k, k++]; AppendTo[t, k], {n, 2, 15}]; t (* Jayanta Basu, Jul 10 2013 *)

PARI

a(n)=my(k=1); while(k!=n*primepi(k),k++); k;
for (n=2, 20, print1(a(n), ", ")); \\ Derek Orr, Aug 13 2014

Python

from math import exp
from sympy import primepi
def a(n):
  m = 2 if n == 2 else int(exp(n)) # pi(m) > m/log(m) for m >= 17
  while m != n*primepi(m): m += 1
  return m
print([a(n) for n in range(2, 10)]) # Michael S. Branicky, Feb 27 2021

Formula

It appears that a(n) is asymptotic to e^2*exp(n). - Chris K. Caldwell, Apr 02 2008

a(n) = A038626(n) * n. - Max Alekseyev, Oct 13 2023

Extensions

Three more terms from Labos Elemer, Sep 12 2003

Edited by N. J. A. Sloane at the suggestion of Chris K. Caldwell, Apr 08 2008

24 terms added and entry a(26) corrected by Jan Büthe, Jan 07 2015