cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A072623 Numbers n such that A065863(n) = 1, i.e., prime(n) mod (n - Pi(n)) = 1.

Original entry on oeis.org

4, 5, 6, 11, 19, 25, 34, 36, 75, 82, 87, 90, 94, 237, 604, 609, 614, 1583, 1592, 10466, 10467, 10498, 10504, 10505, 70501, 70511, 180227, 180294, 180358, 180443, 180447, 466078, 8103422, 21058343, 21058649, 143052872, 143052877, 143053068
Offset: 1

Views

Author

Labos Elemer, Jun 26 2002

Keywords

Comments

A004648, A065134 and A065863 behave similarly; they grow relatively slowly and drop suddenly at unexpected values of n. Parity of A004648 behaves most regularly.
Each cluster of entries exceeds the previous cluster by a power of e.

Examples

			For the cluster started at n = 10466 the remainders of A065863(n) are as follows: {9089, 9092, 9117, 9127, 9148, 9159, 1, 1, 9180, 9183, 9182, 9179, 9172, 9169, 9168, 9177, 9176, 9178, 9183, 9192, 43}. It behaves like A004648 or A065134.

Crossrefs

Cf. A065860-A065863, A065133-A065135, A072608-A072610, A004648, A057809.

Programs

  • Mathematica
    Do[ If[ Mod[ Prime[n], n-PrimePi[n]] == 1, Print[n]], {n, 1, 150000000}]
(* Second program: *)
Position[Table[Mod[Prime[n], n - PrimePi[n]], {n, 10^6}], 1] // Flatten (* Michael De Vlieger, Jul 30 2017 *)

Extensions

Edited by Robert G. Wilson v, Jun 27 2002

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

Views

Author

Labos Elemer, Sep 04 2003

Keywords

Examples

			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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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.

Extensions

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

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

Views

Author

Labos Elemer, Sep 04 2003

Keywords

Links

Crossrefs

Cf. A038623-A038627, A057809, A087235-A087241.

Formula

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

Extensions

More terms from Giovanni Resta, Sep 01 2018

A140174 Numbers k such that k = pi(k)*(sum of the digits of k).

Original entry on oeis.org

0, 2, 30, 1122, 24300
Offset: 1

Views

Author

Farideh Firoozbakht, May 15 2008

Keywords

Comments

This sequence is a subsequence of A057809. There is no further term up to 16*10^6.
a(6) > 1.4*10^22, if it exists. - Giovanni Resta, Aug 31 2018

Examples

			24300 = pi(24300) * (2+4+3+0+0) = 2700 * 9.

Crossrefs

Cf. A000720, A007953, A057809, A140173.

Programs

  • Mathematica
    Select[Range[0,25000],PrimePi[#]Total[IntegerDigits[#]]==#&] (* Harvey P. Dale, Feb 13 2023 *)

A235495 Numbers n such that pi(2n) divides n.

Original entry on oeis.org

1, 2, 3, 4, 48, 50, 60, 504, 540, 546, 558, 561, 564, 567, 4204, 4212, 4220, 4228, 4232, 4236, 32270, 32290, 32305, 32310, 32325, 32345, 32350, 32355, 32360, 240426, 240432, 240504, 240510, 240516, 240522, 240528, 240534, 240540, 240546, 240648, 240678, 240684
Offset: 1

Views

Author

Vincenzo Librandi, Jan 11 2014

Keywords

Examples

			48 is in the sequence because pi(2*48) = 24 and 24 divides 48.

Links

Crossrefs

Cf. A057809.

Programs

  • Mathematica
    Select[Range[10^6],IntegerQ[#/PrimePi[2#]]&]
  • PARI
    is(n)=n%primepi(2*n)==0 \\ Charles R Greathouse IV, Jan 12 2014

A087270 Solutions to gcd(x,pi(x)) = gcd(x, A000720(x)) > 1. Numbers x such that x and pi(x) have common divisor larger than one.

Original entry on oeis.org

4, 6, 8, 10, 14, 15, 16, 20, 22, 24, 27, 30, 33, 38, 39, 40, 44, 46, 48, 50, 51, 54, 56, 58, 62, 63, 64, 66, 72, 75, 77, 78, 80, 82, 90, 92, 93, 94, 96, 100, 102, 105, 108, 114, 115, 116, 117, 118, 120, 122, 123, 124, 125, 126, 132, 134, 136, 138, 140, 142, 144, 146
Offset: 1

Views

Author

Labos Elemer, Sep 16 2003

Keywords

Crossrefs

Cf. A007053, A000720, A087266-A087269, A057809.

Programs

  • Mathematica
    t=Table[GCD[w, PrimePi[w]], {w, 1, 1000}]; Flatten[Position[Sign[t-1], 1]]

A087271 Least number x such that gcd(x, pi(x)) = n.

Original entry on oeis.org

1, 4, 6, 8, 50, 66, 77, 56, 27, 30, 33, 156, 169, 182, 465, 224, 238, 252, 2299, 1380, 189, 902, 207, 96, 100, 1872, 1323, 2464, 1247, 120, 1333, 3168, 528, 1258, 1295, 828, 3441, 2888, 1755, 5800, 1271, 1932, 731, 748, 765, 2852, 2209, 11568, 2695, 4000
Offset: 1

Views

Author

Labos Elemer, Sep 16 2003

Keywords

Examples

			n=253: a(253)=91586, pi(91586)=8855,
gcd(91586, 8855) = 253 first time.

Crossrefs

Cf. A007053, A000720, A087266-A087269, A057809.

Programs

  • Mathematica
    f[x_] := GCD[x, PrimePi[x]]; t=Table[0, {257}]; Do[s=f[n]; If[s<258&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t
Module[{tbl=Table[{x,GCD[x,PrimePi[x]]},{x,12000}]},Table[SelectFirst[ tbl,#[[2]]==n&],{n,50}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 12 2020 *)

Formula

a(n) = Min{x; gcd(x, A000720(x))=n}.

A165689 Numbers n such that pi(n) = (1/10)*n.

Original entry on oeis.org

64540, 64580, 64610, 64620, 64650, 64690, 64700, 64710, 64720
Offset: 1

Views

Author

Farideh Firoozbakht, Oct 06 2009

Keywords

Comments

A subsequence of A057809. For each positive integer m, set of the numbers n such that pi(n)=(1/10^m)*n is a finite set. I guess that all these sets are nonempty. What is the smallest number n such that pi(n) = (1/100)*n?

Crossrefs

Cf. A057809.

Programs

  • Mathematica
    Select[10 Range[10000], PrimePi[ # ] == 1/10 # &]
  • PARI
    forcomposite(n=1, 1e5, if(10*primepi(n) == n, print1(n, ", "))) \\ Altug Alkan, Dec 18 2015
Previous Showing 21-28 of 28 results.

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