A038627 -id:A038627 - cp's OEIS frontend

A057809 Numbers k such that pi(k) divides k.

2, 4, 6, 8, 27, 30, 33, 96, 100, 120, 330, 335, 340, 350, 355, 360, 1008, 1080, 1092, 1116, 1122, 1128, 1134, 3059, 3066, 3073, 3080, 3087, 3094, 8408, 8424, 8440, 8456, 8464, 8472, 23526, 23535, 24300, 64540, 64580, 64610, 64620, 64650, 64690, 64700

Offset: 1

N. J. A. Sloane, Nov 07 2000

nonn

Each cluster of entries is approximately a power of e times larger than the previous cluster.

The sequence is infinite (Golomb, 1962). - Yifan Xie, Jun 23 2025

			120 is a member as there are exactly 30 primes less than 120 and 30 * 4 = 120.

Giovanni Resta, Table of n, a(n) for n = 1..1161 (first 296 terms from Charles R Greathouse IV)
Konstantinos N. Gaitanas, An explicit formula for the prime counting function, arXiv preprint arXiv:1311.1398 [math.NT], 2013.
S. W. Golomb, On the Ratio of N to π(N), The American Mathematical Monthly 69.1 (1962): 36-37.

Cf. A000720, A057810, A071394, A038627, A256394.

Apart from initial term same as A058011.

[n: n in [2..10^5] | n mod #PrimesUpTo(n) eq 0]; // Vincenzo Librandi, Jul 04 2016

select(t -> t mod numtheory:-pi(t) = 0, [$2..10^5]); # Robert Israel, Jul 03 2016

Select[ Range[2, 10^5], IntegerQ[ # / PrimePi[ # ]] & ]
Select[Range[1000], Divisible[#, PrimePi[#]] &] (* Requires version 6.0+. Alonso del Arte, May 24 2015 *)

is(n)=n%primepi(n)==0 \\ Charles R Greathouse IV, Sep 14 2015

More terms from James Sellers, Nov 08 2000

a(297)-a(1161) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

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

A038623 Smallest prime p such that p/pi(p)>=n.

Original entry on oeis.org

2, 2, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401, 554805820711, 1505578023841, 4086199302113

Offset: 1

Jud McCranie

nonn

			pi(37)=12 and a(3)=37 is the smallest prime >= 3*12.

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

Cf. A038624-A038627.
Essentially the same as A062743,A038607.
a(n) = prime(A038624(n)).

Prime[Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]]] (* Jayanta Basu, Jul 10 2013 *)

k=n=1; forprime(p=2,, while(p/k>=n, print1(p", "); n++); k++) \\ Charles R Greathouse IV, Oct 15 2016

a(n) = exp(n + 1 + o(1)). - Charles R Greathouse IV, Oct 15 2016

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar
a(24)-a(28) from David W. Wilson, Apr 25 2017
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A038624 Values of pi(x) where x exceeds n * pi(x).

Original entry on oeis.org

1, 1, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235, 1617175, 4124437, 10553415, 27066974, 69709680, 179992909, 465769803, 1208198526, 3140421716, 8179002096, 21338685407, 55762149030, 145935689361

Offset: 1

Jud McCranie

nonn

"Exceeds" can be interpreted as ">" or ">=" since the corresponding primes are never multiples of their indices. - R. J. Mathar, Jun 08 2008
Equivalently, a(n) = minimal k such that prime(k)/k >= n. - Enoch Haga, Oct 19 2007
a(n) = A062742(n) = A038606(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

			x exceeds 3*pi(x) when pi(x)=12, so a(3)=12

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

Cf. A038623-A038627.
Essentially the same as A062742.
Cf. A038606 (variant).

Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]] (* Jayanta Basu, Jul 10 2013 *)

a(24)-a(28) from Robert G. Wilson v, Sep 26 2005
Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar.
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 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

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

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

A057810 Quotients n/pi(n) for n in A057809.

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane, Nov 07 2000

Giovanni Resta, Table of n, a(n) for n = 1..1161

Cf. A000720, A057809, A038627, A256394.

(#/PrimePi[#] &) /@ Select[Range[2, 3*10^6], IntegerQ[#/PrimePi[#]] &] (* Michael De Vlieger, Apr 15 2015, after N. J. A. Sloane at A057809 *)

More terms from Naohiro Nomoto, Jun 26 2001

A087240 First differences of A087235.

Original entry on oeis.org

25, 87, 240, 774, 1960, 5378, 15828, 40420, 110477, 306255, 823267, 2219935, 6035446, 16314684, 44245201, 119849369, 324308909, 879921037, 2385657333, 6467079803, 17541637367, 47581552613, 129104928784, 350330765356, 950772205549, 2580621276899, 7005302330033

Offset: 2

Labos Elemer, Sep 04 2003

nonn

Giovanni Resta, Table of n, a(n) for n = 2..49

Cf. A038623-A038627, A057809, A087235-A087241.

a(n)=A087235(n+1)-A087235(n)

More terms from Giovanni Resta, Sep 01 2018

cp's OEIS Frontend

A057809 Numbers k such that pi(k) divides k.

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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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A038623 Smallest prime p such that p/pi(p)>=n.

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A038624 Values of pi(x) where x exceeds n * pi(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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A038626 Smallest positive integer m such that m = pi(n*m) = A000720(n*m).

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

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