In connection with [this question](https://mathoverflow.net/questions/283588/how-composite-can-2n-1-be-infinitely-often) and [its follow-up](https://mathoverflow.net/questions/283756/the-limit-superior-of-varphi2n-1-2n-1). 


Suppose that $a\ge 2$ and $b\ne 0$ are integers, and $f$ is a monotonically increasing function such that $f(2)>1$, $f(p)\to\infty$, and the series $\sum\_p 1/f(p)$ (extended onto all rational primes) diverges. Suppose further that for any integer $K>0$, there exist infinitely many integers $n\ge 1$ with $\gcd(a^n+b,K)=1$.



> 
> Is it true that, under the stated assumptions, every point of the interval $[0,1]$ is a limit point of the sequence with the $n$th term
>  $$ \prod\_{p\mid a^n+b}\Big(1-\frac1{f(p)}\Big)? $$
>  In particular, is it true that every point of $[0,1]$ is a limit point of the sequence $\varphi(2^n-1)/(2^n-1)$?
> 
> 
> 



How composite $a^n+b$ is?
This is an answer to the part of the question where the OP is asking if the image of the function 
$$R\_{2,\varphi}: \mathbf N^+ \to \mathbf R: n \mapsto \frac{\varphi(2^n - 1)}{2^n-1}$$ is dense in the interval $[0,1]$. The short answer is yes, some more details follow.




---


This week, I met Carlo Sanna in Turin during a meeting of number theory and mentioned the problem to him. He remembered to have read about the same question in a paper of Florian Luca. Today, I got an e-mail from Carlo with the full reference:



> 
> F. Luca, *On the sum of divisors of the Mersenne numbers*, Math. Slovaca **53**, No. 5 (2003), 457-466 ([EuDML link](https://eudml.org/doc/31788)).
> 
> 
> 


More precisely, see point ii) of the theorem on the bottom of p. 458. 


By the way, Luca points out on p. 459 that:


* his method works for any multiplicative function $f: \mathbf N^+ \to
 \mathbf R$ for which "there exist [...] $c > 0$ and $\lambda > 1$ so
that $f(p^a) = 1 + \frac{c}{p} + O(p^{-\lambda})$ holds for all prime
numbers $p$ and all positive integers $a$", on condition that $[0,1]$ is replaced with the interval $[\liminf\_n f(n), \limsup\_n f(n)]$ (the question in the OP corresponds to the case when $f = R\_{2,\varphi}$);
* the conclusion remains true if the sequence of Mersenne numbers is replaced with a Lucas sequence subjected to some technical conditions.


Among other things, Luca's proof makes use of the [Siegel-Walfisz theorem](https://en.wikipedia.org/wiki/Siegel%E2%80%93Walfisz_theorem).


