Highly composite numbers one less than a (prime) square

Question

I noticed that a surprisingly high share of small highly composite numbers (i.e. positive integers with more divisors than any smaller positive integer) are one less than a prime square:

24   =  5^2-1
48   =  7^2-1
120  = 11^2-1
360  = 19^2-1
840  = 29^2-1
1680 = 41^2-1
5040 = 71^2-1

But then there is a gap: at least the 19 HCNs after 5040 (those listed by Wikipedia) are not p^2-1.

Is this just law of small numbers at work, or is there more to this strange cluster? Are there infinitely many HCNs that are one less than a square?

Answer 1

In the $\ 1000\ $ entries shown in OEIS, the only entry larger than $\ 5040\ $ which is a square minus $\ 1\ $ (and in fact the square of a prime minus $\ 1\ $) , is

$$17297280$$

I do not know whether any other examples exist. Any larger example must exceed $10^{77}$ , hence I do not expect other examples.