A038625 -id:A038625 - cp's OEIS frontend

A087239 First differences of A038625.

25, 69, 234, 678, 2051, 5349, 15118, 41014, 110657, 305655, 823646, 2219386, 6034071, 16316797, 44240660, 119845770, 324311229, 879921169, 2385656018, 6467086046, 17541630385, 47581555131, 129104931215, 350330768077, 950772203169, 2580621278375, 7005302328953

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)=A038625(n+1)-A038625(n)

More terms from Giovanni Resta, Sep 01 2018

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

Original entry on oeis.org

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

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

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

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

A038607 a(n) is the smallest prime number k such that k > n*pi(k), where pi(k) denotes the prime counting function.

Original entry on oeis.org

2, 11, 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, 11091501631019, 30109570413007

Offset: 1

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

a(n) is about exp(n+1+1/(n+1)). - Charles R Greathouse IV, Sep 05 2011

			For n=1, a(1) = 2, since 2 > 1*pi(2) = 1*1. - _N. J. A. Sloane_, Dec 09 2020
For n=3, the 12th prime (37) is the first one satisfying p(k) > 3k.

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

Cf. A038606, A038623.

k = 1; Do[ While[ Prime[k] < n*k, k++ ]; Print[Prime[k]], {n, 1, 25} ]

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

a(n) = prime(A038606(n)) = A000040(A038606(n)).

Extended by Robert G. Wilson v and Ray Chandler, Dec 01 2004
a(26)-a(30) from Charles R Greathouse IV, Sep 05 2011, Sep 06 2011
a(31)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

cp's OEIS Frontend

A087239 First differences of A038625.

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

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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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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).