A038626 Smallest positive integer m such that m = pi(n*m) = A000720(n*m).

1, 9, 24, 66, 168, 437, 1051, 2614, 6454, 15927, 40071, 100346, 251706, 637197, 1617172, 4124436, 10553399, 27066969, 69709679, 179992838, 465769802, 1208198523, 3140421715, 8179002095, 21338685402, 55762149023, 145935689357, 382465573481, 1003652347080, 2636913002890, 6935812012540

Offset: 2

Author

Jud McCranie

Keywords

• nonn

Comments

Golomb shows that solutions exist for each n>1.

For all known terms, we have 2.4*a(n) < a(n+1) < 2.7*a(n) + 7. A038627(n) gives number of natural solutions of the equation m = pi(n*m). - Farideh Firoozbakht, Jan 09 2005

a(n) grows as exp(n)/n. Thus, a(n+1)/a(n) tends to e=exp(1) as n grows. - Max Alekseyev, Oct 15 2017

Examples

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

Links

• Giovanni Resta, Table of n, a(n) for n = 2..50
• 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. A038623, A038624, A038625, A038627, A102281, A087237, A293529.

Formula

a(n) = limit of f^(k)(1) as k grows, where f(x)=A000720(n*x). Also, a(n) = f^(A293529(n))(1). - Max Alekseyev, Oct 11 2017

a(n) = A038625(n) / n. - Max Alekseyev, Oct 13 2023

Extensions

a(24) from Farideh Firoozbakht, Jan 09 2005

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

a(25)-a(32) from Max Alekseyev, Jul 18 2011, Oct 14 2017

a(33)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Aug 31 2018