A038624 -id:A038624 - cp's OEIS frontend

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

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

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

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

A038606 Least k such that k-th prime > n * k.

Original entry on oeis.org

1, 5, 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, 382465573483, 1003652347100

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net) 1998 Jul

nonn

Log(a(n)) =~ -1.295 + 0.964312n. - Robert G. Wilson v, Jan 25 2002
Numbers n such that prime(n) (mod n) begins the next cycle of terms in A004648. Generally prime(i) (mod i) exceeds prime(i-1) (mod i-1) but there are numerous times where for a short run prime(i) (mod i) is minimally less than its predecessor. Here n is substantially less. See Labos's graph.
A090973(a(n)) = n+1. [From Reinhard Zumkeller, Aug 16 2009]
With offset 2: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied. a(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

Giovanni Resta, Table of n, a(n) for n = 1..50
Labos, E. Graph of first 50000 terms of A004648
Andrew R. Booker, The Nth Prime Page

Cf. A038607, A004648, A038625.

A038606 := proc(n)
    for k from 1 do
        if ithprime(k)> n*k then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 24 2013

k = 1; Do[ While[ Floor[ Prime[k]/k] < n, k++ ]; Print[k]; k++, {n, 1, 30} ]

k=1;n=1;forprime(p=3,4e9,if(p/n++>k,print1(n", ");k++)) \\ Charles R Greathouse IV, Sep 06 2011

a(n) = pi(A038607(n)) = A000720(A038607(n)).

Edited by Robert G. Wilson v, Jan 25 2002
a(21)=179992909 corrected by Ray Chandler, Dec 01 2004
a(29)-a(30) from Charles R Greathouse IV, Sep 06 2011
a(31)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062742 Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

Original entry on oeis.org

2, 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

Labos Elemer, Jul 12 2001

nonn

			The q(j)=p(j)/j quotient when the value 14 first appears: {j=251706, p(j)=3523841, q(j)=13.9998291} {251707, 3523901, 14.0000119} {251708, 3523903, 13.9999642} {251709, 3523921, 13.9999801} {251710, 3523957, 14.0000675} {251711, 3523963, 14.0000357}

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

Essentially the same as A038624.

Cf. A038606. - R. J. Mathar, Jan 30 2009

{a062742(m)=local(n,j); for(n=1,m,j=1; while(floor(prime(j)/j)!=n,j++); print1(j,","))} a062742(10^7)

a(n) = Min_{j| floor(p(j)/j) = n}. Note that neither p(j)/j nor floor(p(j)/j) is monotonic.
a(n) = pi(A062743(n)).
a(n) = A038606(n) = A038624(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

More terms from Jason Earls, May 15 2002
a(17)-a(28) from Farideh Firoozbakht and Robert G. Wilson v, Sep 13 2005
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A022465 Numbers n such that prime(n) mod n <= 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 32, 33, 34, 35, 36, 37, 69, 73, 74, 75, 76, 77, 78, 79, 80, 181, 191, 443, 445, 1052, 2701, 2703, 6455, 6456, 6457, 6459, 6460, 6466, 15928, 15929, 16055, 40073, 40078, 40080, 40081, 40082, 40083, 40122

Offset: 1

Clark Kimberling

nonn

a(96) > 5*10^6. - Robert Israel, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..216 (first 95 terms from Robert Israel)

Cf. A038624.

Res:= NULL; p:= 1; count:= 0:
for n from 1 while count < 80 do
  p:= nextprime(p);
  if p mod n <= 10 then count:=count+1; Res:= Res, n;
  fi
od:
Res; # Robert Israel, Aug 29 2018

Select[Range[41000],Mod[Prime[#],#]<=10&] (* Harvey P. Dale, Aug 21 2011 *)

More terms from David W. Wilson

More terms from Robert Israel, Aug 29 2018

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

A090974 Duplicate of A038606.

Original entry on oeis.org

0, 1, 5, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235

Offset: 0

dead

With offset 1: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied for n >= 2. a(n) = A038606(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

cp's OEIS Frontend

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

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A038606 Least k such that k-th prime > n * k.

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A062742 Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

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