A087235 -id:A087235 - cp's OEIS frontend

A087240 First differences of A087235.

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

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

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

A256394 Prime values of pi(n) that divide n.

Original entry on oeis.org

2, 3, 11, 67, 71, 439, 1051, 6469, 40087, 100361, 100363, 251737, 251761, 637319, 637327, 4124459, 10553513, 10553551, 27067277, 69709733, 179993171, 465769817, 3140421769, 8179002109, 8179002133, 55762149029, 55762149071, 382465573489, 1003652347081

Offset: 1

Jonathan Sondow, Apr 13 2015

nonn

a(n) is the largest prime factor of n, since pi(n) ~ n / log n.

			pi(6) = 3 is prime, and 3 divides 6, so 3 is a member.

Giovanni Resta, Table of n, a(n) for n = 1..49
K. Gaitanas, An explicit formula for the prime counting function, arXiv:1311.1398 [math.NT], 2013.
K. Gaitanas, An Explicit Formula for the Prime Counting Function Which is Valid Infinitely Often, Amer. Math. Monthly, 122 (2015), 283.
S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
R. T. Harger and W. L. Hightower, An Interesting Property of x/pi(x), College Math. J., 40 (2009), 213-214.
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A000720, A006530, A038623-A038627, A057809, A057810, A071394, A087235-A087241.

c = 0; lpf[n_] := If[ PrimeQ[n], c++; n, Transpose[ FactorInteger[n]][[1, -1]]]; Do[ If[lpf[n] == c, Print[ PrimePi[n]]], {n, 2, 10^7}]
PrimePi[Select[Select[Range[2,10^6],IntegerQ[#/PrimePi[#]]&],PrimeQ[PrimePi[#]]&]] (* Ivan N. Ianakiev, Apr 15 2015 *)
Select[Table[{PrimePi[n],n},{n,10^6}],PrimeQ[#[[1]]]&&Divisible[#[[2]],#[[1]]]&][[All,1]] (* The program generates the first 9 terms of the sequence. To generate more, increase the constant for n. *) (* Harvey P. Dale, Feb 08 2022 *)

for(n=1,10^6,if(isprime(p=primepi(n))&&!(n%primepi(n)),print1(p,", "))) \\ Derek Orr, Apr 14 2015

a(n) = A000720(A071394(n)) = A006530(A071394(n)).

More terms from Giovanni Resta, Sep 01 2018

A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

Original entry on oeis.org

6, 6, 24, 30, 126, 35, 64, 774, 180, 0, 600, 221, 770, 2145, 32, 4573, 8172, 5852, 5720, 7035, 792, 7774, 5256, 2825, 104, 2484, 1008, 2088, 8880, 9176, 10464, 759, 68, 5880, 23688, 28490, 3420, 49686, 58160, 62074, 136878, 26316, 264, 130320, 16882, 705, 96528, 14063, 95750

Offset: 2

Labos Elemer, Sep 04 2003

nonn

			n=22: a(22) = 10246936436-10246935644 = 792 = 22*36.
a(2) = 6 since x/pi(x) = 2 for x = {2,4,6,8}; 8 - 2 = 6. - _Michael De Vlieger_, Mar 25 2017

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

Last@ # - First@ # & /@ 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) = Max{x; n*pi(n)=x} - Min{x; n*pi(n)=x} = A038625(n) - A087235(n).

a(n) is divisible by n, the quotients are in A087237.

a(27)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

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)

A087239 First differences of A038625.

Original entry on oeis.org

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

A334598 a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n).

Original entry on oeis.org

4, 28, 1860, 149052, 12771496, 1221908916, 132662942122, 16354869261256, 2272946910544740, 353076161059625536, 60799066209732571716, 11518836088596729968092

Offset: 1

Eduard Roure Perdices, May 07 2020

For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
  m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
Fix a real exponent d > 0. If m is big enough, then m < (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
Note that different choices of the exponent d will produce analogous sequences.
The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
  a(13) has 25 digits, starting with 238;
  a(14) has 27 digits, starting with 536;
  a(15) has 30 digits, starting with 1304;
  a(16) has 32 digits, starting with 3409.
The tool primecount [Walisch], used to compute pi(10^28) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.
It is easy to see that for every n >= 1, a(n) is even and a(n)+1 is prime. - Eduard Roure Perdices, Nov 07 2021

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers 18 (2018), Paper No. A61, 14 pp.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal 45 (2018), no. 1, 227-251.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
Kim Walisch, primecount, Github, Aug 14 2021.

Cf. A334599, A087235, A038625.

a(8) from Giovanni Resta, May 07 2020
a(9)-a(10) from Daniel Suteu, May 20 2020
a(11)-a(12) from Eduard Roure Perdices, Nov 07 2021

A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).

Original entry on oeis.org

2, 2, 346, 66942, 7087878, 744600720, 85281842598, 10892966758462, 1553240096780862, 246080334487930558, 43047454015229292840, 8262178422446205100776

Offset: 1

Eduard Roure Perdices, May 07 2020

For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
  m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
Fix a real exponent d > 0. If m is big enough, then m < (m/log(m)) + (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
Note that different choices of the exponent d will produce analogous sequences.
The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
  a(13) has 25 digits, starting with 1729;
  a(14) has 27 digits, starting with 392;
  a(15) has 29 digits, starting with 962;
  a(16) has 32 digits, starting with 2534.
The tool primecount [Walisch], used to compute pi(10^28) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.
It is easy to see that for every n >= 1, a(n) is even and a(n)+1 is prime. - Eduard Roure Perdices, Nov 07 2021

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers 18 (2018), Paper No. A61, 14 pp.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal 45 (2018), no. 1, 227-251.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.
Kim Walisch, primecount, Github, Aug 14 2021.

Cf. A334598, A087235, A038625.

a(8) from Giovanni Resta, May 07 2020
a(9)-a(10) from Daniel Suteu, May 20 2020
a(11)-a(12) from Eduard Roure Perdices, Nov 07 2021

cp's OEIS Frontend

A087240 First differences of A087235.

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

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A256394 Prime values of pi(n) that divide n.

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A087236 a(n) is the difference between the largest and smallest integer solutions to n=x/pi(x), where pi(x) = A000720(x).

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A087238 First differences of A057809.

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A087239 First differences of A038625.

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A334598 a(n) is the largest nonnegative integer m such that m >= pi(m)^(1 + 1/n).

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A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).

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