OFFSET
1,1
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
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.
EXAMPLE
The values of phi(p+1)/phi(p-1) for the first terms are 1 < 1.031... < 1.06 < 1.118... < 1.12 < ...
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 11 2019
EXTENSIONS
a(24)-a(28) from Giovanni Resta, Nov 01 2019
STATUS
approved