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

2,2

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

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.

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

Example

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

Crossrefs

Cf. A038623, A038624, A038625, A038627, A102281, A087237, A293529.

Sequence in context: A079771 A297225 A258438 * A195970 A223372 A180334

Adjacent sequences: A038623 A038624 A038625 * A038627 A038628 A038629

Keyword

nonn

Author

Jud McCranie

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

Status

approved