A038627 -id:A038627 - cp's OEIS frontend

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

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

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

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

cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A086511 a(n) is the smallest integer k > 1 such that k > n * pi(k), where pi() denotes the prime counting function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A087238 First differences of A057809.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A087239 First differences of A038625.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions