A087237 -id:A087237 - cp's OEIS frontend

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

Offset: 1

Jud McCranie

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

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

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.

Cf. A038623, A038624, A038625, A038626, A102281, A087237.

(* 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;
  xmin = Infinity; xmax = 0; While[x <= x1,
   If[x == PrimePi[n x], c++; xmin = Min[x, xmin];
xmax = Max[x, xmax]]; x++]; c, {n, 2, 15}]] (* Robert Price, Mar 28 2020 *)

One more term from Labos Elemer, Sep 05 2003
a(24)-a(26) from Labos Elemer, Sep 12 2003
Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar
a(27)-a(29) from David Radcliffe, Sep 10 2014
a(29) corrected and a(30)-a(50) obtained from the A038625 values computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

A102281 a(n) is the largest number m such that m = pi(n*m).

Original entry on oeis.org

4, 11, 30, 72, 189, 442, 1059, 2700, 6472, 15927, 40121, 100363, 251761, 637340, 1617174, 4124705, 10553853, 27067277, 69709965, 179993173, 465769838, 1208198861, 3140421934, 8179002208, 21338685406, 55762149115, 145935689393

Offset: 2

Farideh Firoozbakht, Jan 09 2005; extended Sep 13 2005

nonn

All known terms of this sequence satisfy the relation 2.4*a(n) - 12 < a(n+1) < 2.7*a(n) + 1 is true.

a(n) is the largest number m such that floor(prime(m)/m)=n-1. - Farideh Firoozbakht, Sep 13 2005

			3140421934 = pi(24*3140421934) and 3140421934 is the largest number with this property, so a(24) = 3140421934.

Giovanni Resta, Table of n, a(n) for n = 2..50

Cf. A038625, A038626, A038627, A087237.

a(24) corrected by Max Alekseyev, Jul 18 2011

a(29)-a(50) obtained from the A038625 values computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

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

Original entry on oeis.org

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

Jud McCranie

nonn

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

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

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.

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

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

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

A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

Original entry on oeis.org

6, 6, 24, 30, 126, 35, 64, 774, 180, 0, 600, 221, 770, 2145, 32, 4573, 8172, 5852, 5720, 7035, 792, 7774, 5256, 2825, 104, 2484, 1008, 2088, 8880, 9176, 10464, 759, 68, 5880, 23688, 28490, 3420, 49686, 58160, 62074, 136878, 26316, 264, 130320, 16882, 705, 96528, 14063, 95750

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			n=22: a(22) = 10246936436-10246935644 = 792 = 22*36.
a(2) = 6 since x/pi(x) = 2 for x = {2,4,6,8}; 8 - 2 = 6. - _Michael De Vlieger_, Mar 25 2017

Cf. A038623-A038627, A057809, A087235-A087240.

Last@ # - First@ # & /@ Values@ Rest@ KeySort@ PositionIndex@ Table[n/PrimePi[n] /. k_ /; Not@ IntegerQ@ k -> 0, {n, 2, 10^6}] (* Michael De Vlieger, Mar 25 2017, Version 10 *)

a(n) = Max{x; n*pi(n)=x} - Min{x; n*pi(n)=x} = A038625(n) - A087235(n).

a(n) is divisible by n, the quotients are in A087237.

a(27)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

cp's OEIS Frontend

A038627 Number of solutions x to n * pi(x) = x, where pi(x) = number of primes <= x.

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A102281 a(n) is the largest number m such that m = pi(n*m).

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

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A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

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Mathematica

Formula

Extensions

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