We continue from Queries on Twin Primes - 1.
Qn: There are instances of "highly composites between twin primes" such as 60 which falls between twin primes 59 and 61 and 180 which is between 179 and 181. Can an upper bound be established for such highly composite numbers?
Heuristically, we should expect only finitely many such numbers. If $C(x)$ is the number of highly composite numbers which are less than or equal to $x$, then Erdos and Nicholas showed that there are positive constants $a$ $b$, $K_1$ and $K_2$, such such that $1 < a< b$ and $K_1 (\log x)^a < C(x) < K_2(\log x)^b $ . So the $n$th highly composite number grows at least as fast as $ e^{n^T}$ for some constant $T>1$. Then by the prime number theorem, the chance that we have two twin primes on either side should be roughly $$\left(\frac{1}{\log e^{n^T}}\right)^2 = \frac{1}{n^{2T}}.$$
But $\sum_{n=1}^{\infty} \frac{1}{n^{2T}}$ converges so we expect there to be only finitely many $n$ that work.
This heuristic may not be completely persuasive because highly composite numbers have a lot of prime factors, so numbers 1 away from them should have a slightly higher chance of being prime than one would randomly expect. However, this should alter things only by a small factor, likely not enough to change the overall behavior.
