A038627 - OEIS

A038627

Number of solutions x to n * pi(x) = x, where pi(x) = number of primes <= x.

0, 4, 3, 3, 6, 7, 6, 6, 3, 9, 1, 18, 11, 12, 21, 3, 10, 33, 31, 32, 24, 8, 13, 32, 35, 4, 15, 9, 15, 26, 22, 24, 9, 3, 14, 55, 36, 3, 65, 52, 33, 139, 42, 2, 85, 25, 7, 96, 16, 33

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

OFFSET

1,2

COMMENTS

Equivalently, a(n) is number of solutions x to the equation pi(n*x) = x. - Farideh Firoozbakht, Jan 09 2005 [For example, a(2) = 4 because 1, 2, 3 & 4 are all solutions of pi(2*x) = x and a(11) = 1 because 15927 is the only solution of the equation pi(11*x) = x.]

S. W. Golomb proved that a(n) > 0 for each integer n > 1. - Carlo Sanna, Nov 09 2015

LINKS

Table of n, a(n) for n=1..50.

S. W. Golomb, On the ratio of N to π(N), The American Mathematical Monthly, Vol. 69, No. 1 (Jan 1962), pp. 36-37.

Robert T. Harger and William L. Hightower, An Interesting Property of x/π(x), The College Mathematics Journal Vol. 40, No. 3 (May 2009), pp. 213-214.

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

EXAMPLE

11*pi(x) = x has only 1 solution, so a(11) = 1.

MATHEMATICA

(* Assumes upper and lower bounds are as defined in A038626. *)

xmin = .5; xmax = 2;

Join[{0}, Table[c = 0; x = Floor[2.4*xmin]; x1 = 2.7*xmax + 7;