Prime numbers which end with $59$ or $79$

Question

This is the weak conjecture of https://math.stackexchange.com/questions/4834936/prime-numbers-which-end-with-19-39-59-79-or-99.\ $\varphi(n)$ denotes the Euler’s totient function, $n$ denotes a natural number $> 1$ and $\sigma(n)$ is the sum of the divisors of $n$. Consider the expression: $$F(n)=\varphi(\sigma(\sigma(n))-1)+1$$

Conjecture: if this ends with 59 or 79 then this number is always prime.

I checked my conjecture up to $n=10^7$ with the help of a member (see the previous topic).

What do you think?

Answer 1

It's still not true. There is

• $n = 29194758, F(n) = 178136579 = 137^2 \cdot 9491$
• $n = 113547153, F(n) = 409620359 = 16097 \cdot 25447$

These are the smallest examples found by a simple brute-force search in Sage, using the code I linked in my comment on your previous post.