Do highly composite numbers of the form n^2-1 occur infinitely often? What about when n is prime? Most values of n for which this holds seem to be. (Which makes sense, because if n is highly composite, n+1 is going to be deficient (because n’s prime factors will all go away), and squares of primes certainly fit the bill).

(Actually, can we always say that a highly composite number + 1 is either prime or the square of a prime? All of the values I checked were.

Things start to change around 5040…)