Happy numbers led me to Fortunate numbers, which are Fortunate not because they are well off but because the guy studying them was named Fortune. https://en.wikipedia.org/wiki/Fortunate_number
Fortune conjectured that all Fortunate numbers are prime. I know jack about number theory but isn't that obvious from the density of primes? If there were a composite Fortunate number Fn it couldn't have any factors less than the nth prime number, which means it would have to be greater than the nth prime number squared.
Not at all obvious. It is a very open question whether prime gaps are bounded by ~(log N)^2... AFAIK, even assuming the GRH, the best upper bound is roughly N^(1/2).