This is a curiosity question.
Let $N$ be positive integer, I just want to know how many (an approximation) pair of primes $(p_1,p_2)$ that are less than $n$ and verify the following identity: $$\tau(p_1-1)=\tau(p_2-1) $$ where $\tau(n)$ stands for the number of divisors of $n$. more formally :
For a large integer $N$, is there an approximation of the following number: $$F(N)=\left| \left\{(p_1,p_2)\in \mathbb P_N^2\big /p_1\neq p_2 \ \text{ and } \tau(p_1-1)=\tau(p_2-1)\right\}\right| $$ where $\mathbb P_N $ is the set of primes less than or equal to $N$.
My attempts Let's just evaluate $F$ at primes: $F(2)=0,F(3)=0,F(5)=0,F(7)=0,F(11)=2, F(13)=2,F(19)=4,...$
Thank you.