A038626 -id:A038626 - cp's OEIS frontend

A293529 Smallest number of iterations of f(x)=A000720(n*x) at x=1 to reach the limit (=A038626(n)).

0, 8, 12, 18, 26, 39, 48, 53, 63, 76, 117, 136, 126, 186, 179, 217, 251, 274, 321, 357, 355, 391, 469, 506, 515, 620, 623, 713, 722, 771, 898

Offset: 2

Max Alekseyev, Oct 11 2017

Cf. A000720, A038626.

a(n) = smallest k such that f^(k)(1)=A038626(n).

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

Original entry on oeis.org

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

Jud McCranie

nonn

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

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

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.

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

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

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

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

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

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

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

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

Original entry on oeis.org

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

A087235 a(n) is the largest number in the set of solutions to n=x/pi(x), where pi(x)=A000720(x).

Original entry on oeis.org

8, 33, 120, 360, 1134, 3094, 8472, 24300, 64720, 175197, 481452, 1304719, 3524654, 9560100, 25874784, 70119985, 189969354, 514278263, 1394199300, 3779856633, 10246936436, 27788573803, 75370126416, 204475055200, 554805820556, 1505578026105, 4086199303004, 11091501633037

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			n=22: list of solutions = {10246935644, 10246935842, 10246935864, 10246935974, 10246936106, 10246936128, 10246936370, 10246936436}, so a(22)=10246936436.

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

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

a(n) = Max{x; n*pi(x)=x}.

More terms from David Radcliffe, Sep 10 2014

a(29) corrected and a(30)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 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

A087237 a(n) = (Max{x : npi(x) = x} - Min{x : npi(x) = x})/n = A087236(n)/n.

Original entry on oeis.org

3, 2, 6, 6, 21, 5, 8, 86, 18, 0, 50, 17, 55, 143, 2, 269, 454, 308, 286, 335, 36, 338, 219, 113, 4, 92, 36, 72, 296, 296, 327, 23, 2, 168, 658, 770, 90, 1274, 1454, 1514, 3259, 612, 6, 2896, 367, 15, 2011, 287, 1915

Offset: 2

Labos Elemer, Sep 04 2003

nonn

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

Equivalently, a(n) is the difference between largest solution and smallest natural solution of the equation pi(n*x) = x. - Farideh Firoozbakht, Jan 09 2005 [Max Alekseyev observes that this is trivially equivalent to the first definition.]

			n=22: a(22) = (10246936436-10246935644)/22 = 792/22 = 36.
a(2) = 3 because 1, 2, 3 & 4 are all solutions of pi(2*x) = x and 4 - 1 = 3; a(11) = 0 because 15927 is the only solution of the equation pi(11*x) = x and 15927 - 15927 = 0. - _Farideh Firoozbakht_, Jan 09 2005

Jan Büthe, Table of n, a(n) for n = 2..50

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

Cf. A102281.

MapIndexed[(Last@ #1 - First@ #1)/(First@ #2 + 1) &, 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) = A102281(n) - A038626(n). - Farideh Firoozbakht, Jan 09 2005

Edited by N. J. A. Sloane, Oct 28 2008 at the suggestion of R. J. Mathar

24 additional terms from Jan Büthe, Jan 16 2015

A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

Original entry on oeis.org

2, 9, 28, 121, 336, 1081, 3060, 8409, 23527, 64541, 175198, 480865, 1304499, 3523885, 9557956, 25874753, 70115413, 189961183, 514272412, 1394193581, 3779849620, 10246935645, 27788566030, 75370121161, 204475052376, 554805820453, 1505578023622, 4086199301997

Offset: 1

Tim Paulden (timmy(AT)cantab.net), Sep 09 2003

nonn

a(n) is bounded above by the sequence A038623, in which k is required to be prime. In addition, the sequence pi(a(n)) = {1, 4, 9, 30, 67, 180, 437, 1051, ...} closely resembles the sequence A038624, in which the n-th term is the minimal t such that k >= n * pi(k) for every k satisfying pi(k) = t. If we were to make the inequality in A038624 strict, the resulting sequence would provide an upper bound for pi(a(n)). Sequences A038625, A038626 and A038627 focus on the equality k = n * pi(k): as we would expect, a(n) follows A038625 very closely for large n.

			Consider the pairs (k, pi(k)) for k > 1. The inequality k > 1 * pi(k) is first satisfied at k = 2 and so a(1) = 2. Similarly, the inequality k > 2 * pi(k) is first satisfied at k = 9 and so a(2) = 9.

Giovanni Resta, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A038623, A038624, A038625, A038626, A038627.

a(n) = { k = 2; while (k <= n*primepi(k), k++); return (k);} \\ Michel Marcus, Jun 19 2013

Heuristically, for large n, a(n) ~= 3.0787*(2.70888^n) [error < 0.05% for 15 <= n <= 20].
From Nathaniel Johnston, Apr 10 2011: (Start)
a(n) >= exp(n/2 + sqrt(n^2 + 4n)/2), n >= 6.
a(n) = A038625(n) + m(n)*n + 1 for some m(n) >= 0. For n = 2, 3, 4, ..., m(n) = 3, 0, 6, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
(End)

a(21)-a(26) from Nathaniel Johnston, Apr 10 2011
Corrected a(26) and a(27)-a(28) from Giovanni Resta, Sep 01 2018
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

cp's OEIS Frontend

A293529 Smallest number of iterations of f(x)=A000720(n*x) at x=1 to reach the limit (=A038626(n)).

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

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

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A087235 a(n) is the largest number in the set of solutions to n=x/pi(x), where pi(x)=A000720(x).

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

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A087237 a(n) = (Max{x : n*pi(x) = x} - Min{x : n*pi(x) = x})/n = A087236(n)/n.

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A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

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PARI

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Extensions

A087237 a(n) = (Max{x : npi(x) = x} - Min{x : npi(x) = x})/n = A087236(n)/n.