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.

Showing 1-6 of 6 results.

A066902 Integers k such that phi(prime(k)+1) = phi(prime(k)-1).

Original entry on oeis.org

3, 5, 20, 156, 254, 377, 593, 1800, 5903, 5981, 7925, 18669, 19240, 41274, 48296, 135700, 146866, 228028, 234303, 251216, 407377, 654288, 802222, 886223, 938654, 1063412, 1072766, 1212140, 1238668, 1515063, 1609346, 2080991, 2097725, 2363130, 2408674, 2916514
Offset: 1

Views

Author

Benoit Cloitre, Jan 26 2002

Keywords

Comments

Integers k such that A008331(k) = A008330(k).

Links

Crossrefs

Cf. A000010, A008864, A006093, A008330, A008331, A067890.

Programs

  • Mathematica
    Select[Range[1000000],EulerPhi[Prime[#]-1]==EulerPhi[Prime[#]+1]&] (* Harvey P. Dale, Feb 25 2012 *)
  • PARI
    isok(k) = my(p=prime(k)); eulerphi(p+1) == eulerphi(p-1); \\ Michel Marcus, Apr 06 2020

Formula

a(n) = primepi(A067890(n)). - Giovanni Resta, Apr 06 2020

Extensions

More terms from Harvey P. Dale, Feb 25 2012
More terms from Jinyuan Wang, Apr 05 2020

A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

Original entry on oeis.org

-1, 1, 5, 8, 14, 22, 25, 31, 28, 48, 56, 73, 78, 76, 56, 80, 74, 138, 112, 120, 159, 136, 102, 156, 210, 185, 168, 126, 240, 212, 248, 212, 226, 240, 226, 300, 314, 283, 204, 252, 222, 474, 296, 412, 339, 388, 472, 360, 270, 472, 378, 368, 634, 396, 427, 316, 404, 592, 534, 628, 436, 434, 582, 480, 684, 456, 700, 836
Offset: 1

Views

Author

Labos Elemer, Mar 03 2004

Keywords

Comments

The sequence differs from A065394 since it is not monotonic.

Examples

			a(1) = sigma(phi(2))- phi(sigma(2)) = sigma(1)-phi(3) = 1-2 = -1.

Links

Crossrefs

Cf. A000010, A000040, A000203, A033632, A065393, A065394, A065395, A008331, A008332, A092584, A092585, A092586, A092587, A092588, A092589.

Programs

  • Magma
    [DivisorSigma(1,EulerPhi(p))-EulerPhi(DivisorSigma(1,p)): p in PrimesUpTo(400)]; // Bruno Berselli, Oct 20 2015
  • Mathematica
    Table[DivisorSigma[1, p-1] - EulerPhi[p+1], {p, Prime[Range[100]]}] (* Amiram Eldar, Jun 09 2024 *)

Formula

a(n) = sigma(prime(n)-1) - phi(prime(n)+1) = A008332(n) - A008331(n). - Amiram Eldar, Jun 09 2024

A067732 Numbers k such that phi(prime(k) + 1) == 0 (mod k).

Original entry on oeis.org

1, 2, 4, 6, 8, 24, 36, 48, 80, 192, 1440, 3960, 5184, 5478, 6480, 16080, 20336, 29628, 40073, 40083, 43056, 44980, 85800, 200320, 241920, 252378, 258048, 287040, 697248, 1805672, 3669120, 3713787, 3927600, 4321143, 4448880, 6928128
Offset: 1

Views

Author

Benoit Cloitre, Feb 05 2002

Keywords

Links

Crossrefs

Cf. A008331.

Programs

  • Mathematica
    Do[ If[ Mod[ EulerPhi[ Prime[n] + 1], n] == 0, Print[n]], {n, 1, 10^7} ]
  • PARI
    isok(k) = (eulerphi(prime(k)+1) % k) == 0; \\ Michel Marcus, Feb 17 2021
  • PARI
    list(lim) = my(k = 0); forprime(p = 1, lim, k++; if(!(eulerphi(p+1) % k), print1(k, ", "))); \\ Amiram Eldar, Apr 25 2025

Extensions

Edited and extended by Robert G. Wilson v, Feb 07 2002

A326356 Lesser of twin primes p >= 5 for which phi(p+1)/phi(p-1) reaches record value, where phi(n) is the Euler totient function (A000010).

Original entry on oeis.org

5, 2381, 3851, 20021, 50051, 52361, 424271, 470471, 602141, 2302301, 6806801, 16926911, 17497481, 69989921, 78278201, 183953771, 242662421, 468818351, 2156564411, 24912037151, 43874931101, 73769375681, 131104243271, 1360122864101, 1943064533411, 2635321709021, 3075260848661, 4078063299311
Offset: 1

Views

Author

Amiram Eldar, Sep 11 2019

Keywords

Comments

Terms a(2)-a(23) were taken from the paper by Garcia et al.
Garcia et al. proved that assuming Dickson's conjecture, {phi(p+1)/phi(p-1) : p and p+2 are prime} is dense in [0, oo), and thus this sequence is infinite.
They give an example of a term p with 1099 digits with phi(p+1)/phi(p-1) = 3.11615...
What is the least value of lesser of twin primes p such that phi(p+1)/phi(p-1) > 2?
A candidate is p = 8183287190196092135163947564054981234789530779544672356881 for which the ratio is equal to 2.00047615... . - Giovanni Resta, Nov 01 2019

Examples

			The values of phi(p+1)/phi(p-1) for the first terms are 1 < 1.031... < 1.06 < 1.118... < 1.12 < ...

Links

Crossrefs

Cf. A000010, A001359, A006512, A008330, A008331, A286714, A303549, A326391.
Except for 5, subsequence of A286715.

Programs

  • Mathematica
    s = {}; rm = 0; p = 5; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = EulerPhi[p + 1]/EulerPhi[p - 1]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^6}]; s

Extensions

a(24)-a(28) from Giovanni Resta, Nov 01 2019

A363700 a(n) = phi(2*prime(n)+1).

Original entry on oeis.org

4, 6, 10, 8, 22, 18, 24, 24, 46, 58, 36, 40, 82, 56, 72, 106, 96, 80, 72, 120, 84, 104, 166, 178, 96, 168, 132, 168, 144, 226, 128, 262, 200, 180, 264, 200, 144, 216, 264, 346, 358, 220, 382, 252, 312, 216, 276, 296, 288, 288, 466, 478, 264, 502, 408, 480, 420, 360, 288, 562
Offset: 1

Views

Author

Alain Rocchelli, Jun 16 2023

Keywords

Comments

2*prime(n)+1 is prime iff a(n) = 2*prime(n).

Crossrefs

Cf. A000010, A000040, A008331, A037225, A072055.

Programs

  • Mathematica
    a[n_] := EulerPhi[2*Prime[n] + 1]; Array[a, 100] (* Amiram Eldar, Jun 16 2023 *)
  • PARI
    a(n)=eulerphi(2*prime(n)+1)

Formula

a(n) = A000010(A072055(n)).
a(n) = A037225(A000040(n)).

A375337 a(n) = binomial(prime(n), phi(prime(n) + 1)).

Original entry on oeis.org

1, 3, 10, 35, 330, 1716, 12376, 75582, 490314, 4292145, 300540195, 17672631900, 7898654920, 960566918220, 1503232609098, 64617565719070, 109712808959985, 232714176627630544, 13413576695470557606, 5300174441392685400, 873065282167813104916, 13146145590943010676030
Offset: 1

Views

Author

Mike Jones, Aug 12 2024

Keywords

Examples

			a(4) = 35 because binomial(prime(4), phi(prime(4) + 1)) = binomial(7, phi(8)) = binomial(7, 4) = 35.

Crossrefs

Cf. A007318, A008331.

Programs

  • Mathematica
    Map[Binomial[#, EulerPhi[# + 1]] &, Prime[Range[22]]] (* Amiram Eldar, Aug 13 2024 *)

Formula

a(n) << 2^p/sqrt(p), where p = prime(n). - Charles R Greathouse IV, Aug 12 2024
Showing 1-6 of 6 results.

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