A057809 -id:A057809 - cp's OEIS frontend

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

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

A087241 Consecutive min and max-terms of solution-clusters of A057809, i,e, least and largest solutions to n=x/A000720[x].

Original entry on oeis.org

2, 8, 27, 33, 96, 120, 330, 360, 1008, 1134, 3059, 3094, 8408, 8472, 23526, 24300, 64540, 64720, 175197, 175197, 480852, 481452, 1304498, 1304719, 3523884, 3524654, 9557955, 9560100, 25874752, 25874784, 70115412, 70119985, 189961182, 189969354, 514272411, 514278263

Offset: 2

Labos Elemer, Sep 04 2003

nonn

Giovanni Resta, Table of n, a(n) for n = 2..99

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

More terms from Giovanni Resta, Sep 01 2018

A071394 Numbers n divisible by pi(n) [A057809] with prime pi(n); i.e., largest prime factor of n equals pi(n).

Original entry on oeis.org

4, 6, 33, 335, 355, 3073, 8408, 64690, 481044, 1304693, 1304719, 3524318, 3524654, 9559785, 9559905, 70115803, 189963234, 189963918, 514278263, 1394194660, 3779856591, 10246935974, 75370122456, 204475052725, 204475053325, 1505578023783, 1505578024917

Offset: 1

Jason Earls, Jun 12 2002

nonn

			pi(8408) = 1051 and 8408 = 2*2*2*1051.

Giovanni Resta, Table of n, a(n) for n = 1..49

Cf. A000720, A006530, A256394.

c = 0; lpf[n_] := If[ PrimeQ[n], c++; n, Transpose[ FactorInteger[n]][[1, -1]]]; Do[ If[ lpf[n] == c, Print[n]], {n, 2, 10^7}]
Select[Select[Range[2,10^6],IntegerQ[#/PrimePi[#]]&],PrimeQ[PrimePi[#]]&] (* Ivan N. Ianakiev, Apr 15 2015 *)
Select[Range[10^6], FactorInteger[#][[-1, 1]] == PrimePi@ # &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = isprime(p=primepi(n)) && !(n % p); \\ Michel Marcus, Jul 31 2017

A000720(a(n)) = A006530(a(n)) = A256394(n). - Jonathan Sondow, Apr 15 2015

Edited and extended by Robert G. Wilson v, Jun 13 2002
More terms from Hans Havermann, Jul 02 2002
a(26)-a(27) from Giovanni Resta, Mar 28 2017

A087238 First differences of A057809.

Original entry on oeis.org

2, 2, 2, 19, 3, 3, 63, 4, 20, 210, 5, 5, 10, 5, 5, 648, 72, 12, 24, 6, 6, 6, 1925, 7, 7, 7, 7, 7, 5314, 16, 16, 16, 8, 8, 15054, 9, 765, 40240, 40, 30, 10, 30, 40, 10, 10, 10, 110477, 305655, 12, 144, 12, 12, 12, 12, 12, 12, 12, 204, 60, 12, 36, 12, 12, 12, 12, 823046, 13

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			Large differences arise between max term in a cluster and least one in the next cluster of A057809. While small entries obtained as differences inside a cluster.
E.g.:...,21,6467079011,198,22,110,132,22,242,66,17541629593,23,... shows by first differences the transition from the 21st cluster to 23rd solution-set over the 22nd-set with multiples of 22.
First cluster is empty, while 11th contains one term (see A038227).

Giovanni Resta, Table of n, a(n) for n = 2..1161

Cf. A038623-A038627, A057809, A087235-A087242.

Differences[Select[Range[2,1500000],Divisible[#,PrimePi[#]]&]] (* Harvey P. Dale, Aug 13 2012 *)

a(n)=A057809(n+1)-A057809(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

A065134 Remainder when n is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19

Offset: 2

Labos Elemer, Oct 15 2001

nonn

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)

{ for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = n (mod pi(n)).

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009

Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

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

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 9, 2, 8, 10, 14, 22, 3, 9, 15, 19, 23, 29, 41, 39, 63, 69, 2, 6, 16, 16, 24, 42, 48, 52, 54, 52, 74, 84, 88, 102, 114, 122, 134, 152, 156, 166, 168, 1, 7, 13, 19, 23, 31, 71, 71, 73, 73, 65, 77, 91, 79, 91, 109, 115, 125, 137, 149, 155, 185, 197, 203, 197, 235

Offset: 1

Labos Elemer, Oct 08 2002

			a(4) = 3 since prime(prime(4)) (mod prime(4)) = prime(7) (mod 7) = 17 (mod 7) = 3. - _Michael De Vlieger_, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076241, A076242, A076243.

a:= n-> (p-> irem(ithprime(p), p))(ithprime(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 09 2015

Table[Mod @@ Map[Nest[Prime, n, #] &, {2, 1}], {n, 65}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = prime(prime(n)) % prime(n); \\ Michel Marcus, Mar 25 2017

a(n) = prime^2(n) mod prime(n) = A006450(n) mod A000040(n).

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

A073436 Smallest k such that k mod pi(k) = n.

Original entry on oeis.org

2, 3, 5, 7, 16, 21, 22, 25, 26, 29, 32, 65, 66, 70, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 99, 106, 116, 117, 118, 119, 218, 219, 220, 221, 222, 247, 248, 249, 250, 255, 256, 261, 262, 267, 268, 289, 290, 291, 292, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n) > a(n-1) except for 68, 180, 1051, 6454, 6456, 6459, 40073, 40078, ..., . - Robert G. Wilson v, Feb 24 2023

			Remainder 7 appears first as 25 mod pi(25) = 25 mod 9 = 7, so a(7) = 25.

Giovanni Resta, Table of n, a(n) for n = 0..10000

Cf. A000720, A038625, A057809, A065134, A073435, A073437, A360789.

with(numtheory); f:=proc(n) local i,j,k; for i from 2 to 10000 do if i mod pi(i) = n then RETURN(i); fi; od: RETURN(-1); end; # N. J. A. Sloane, Sep 01 2008

a = Compile[{{n, Integer}}, Block[{k = 2}, While[ Mod[k, PrimePi@ k] != n, k++]; k]]; Array[a, 59, 0] (* _Robert G. Wilson v, Feb 24 2023 *)

a(n)={my(q=0, k=2);forprime(p=3, oo, q++; while(kAndrew Howroyd, Feb 23 2023

a(n) = Min{k: k mod A000720(k) = n} = Min{k: A065134(k) = n}.

a(0) from Robert G. Wilson v, Feb 23 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A087241 Consecutive min and max-terms of solution-clusters of A057809, i,e, least and largest solutions to n=x/A000720[x].

Views

Author

Keywords

Links

Crossrefs

Extensions

A071394 Numbers n divisible by pi(n) [A057809] with prime pi(n); i.e., largest prime factor of n equals pi(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A087238 First differences of A057809.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A065134 Remainder when n is divided by the number of primes not exceeding n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

Views

Author

Keywords

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