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 1-3 of 3 results.

A286715 Lesser of twin primes for which phi(p-1) < phi(p+1).

Original entry on oeis.org

3, 2381, 3851, 14561, 17291, 20021, 20231, 26951, 34511, 41231, 47741, 50051, 52361, 55931, 57191, 65171, 67211, 67271, 70841, 82811, 87011, 98561, 101501, 101531, 108461, 117041, 119771, 126491, 129221, 134681, 136991, 142871, 145601, 150221, 156941, 165551, 166601
Offset: 1

Views

Author

Michel Marcus, May 13 2017

Keywords

Links

Crossrefs

Cf. A001359, A006512, A008330, A286714.

Programs

  • Mathematica
    s = {}; p = 3; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; If[EulerPhi[p + 1] > EulerPhi[p - 1], AppendTo[s, p]]; p = q, {15500}]; s (* Amiram Eldar, Sep 11 2019 *)
  • PARI
    lista(nn) = forprime(p=2, nn, if (isprime(p+2) && (eulerphi(p-1) < eulerphi(p+1)), print1(p, ", ")));

A303549 Lesser of twin primes p for which phi(p-1) = phi(p+1), where phi(n) is the Euler totient function (A000010).

Original entry on oeis.org

5, 11, 71, 2591, 208391, 16692551, 48502931, 92012201, 249206231, 419445251, 496978301, 1329067391, 1837151681, 2277479051, 2647600061, 4733566391, 6435087011, 10327948751, 14089345691, 14923624031, 22415286251, 27508270301, 39662281331, 59013882071, 70353395351
Offset: 1

Views

Author

Amiram Eldar, Apr 26 2018

Keywords

Comments

Intersection of A001359 and A067890 (or A066812).
The terms below 10^8 were taken from the paper by Garcia et al.

Examples

			p = 5 is the lesser of the twin primes (5, 7), and phi(5-1) = phi(5+1) = 2.

Links

Crossrefs

Cf. A000010, A001359, A006512, A008330, A066812, A067890, A067933, A286714, A286715.

Programs

  • Mathematica
    seq={}; Do[p = Prime[i]; If[PrimeQ[p+2] && EulerPhi[p-1] == EulerPhi[p+1], AppendTo[seq, p]], {i, 1, 1000000}]; seq
  • PARI
    isok(p) = isprime(p) && isprime(p+2) && (eulerphi(p-1) == eulerphi(p+1)); \\ Michel Marcus, Apr 26 2018

Extensions

a(12)-a(16) from Michel Marcus, Apr 26 2018
a(17)-a(25) from Giovanni Resta, Apr 26 2018

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
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