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).

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

Author

Amiram Eldar, Sep 11 2019

Keywords

• nonn

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

• Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe Udell, Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function, arXiv:1906.05927 [math.NT], 2019.
• Wikipedia, Dickson's conjecture.

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