Richey Numbers: Update

Recently, I discovered a strange list of primes while looking at the prime factorizations of Richey Numbers. (Dated January 3rd.) I the primes arose in the prime factorizations of Richey Numbers, and only those primes appeared in the factorizations. I realized that the list is all the primes of the form 4m+1, for m>0. I therefore conjectured that any number of the form 5(4m+1) = 20m+5 is a Richey Number when m is prime. (The 5 is there because the primes appeared in the factorizations attached always to a 5. For example, 65 = 5*13, 85 = 5*17, and so on.)

This finding also accounts for numbers of the form (4m+1)^2 = 16m^2+8m+1 and (4m+1)(4k+1), for 4k+1 and 4m+1 both prime. However, I haven't yet accounted for one strange kind of Richey Number; those of the form 5^n for n>2, such as 125, 625 and 3125.

I haven't algebraically shown why this is true yet, and I'm not really sure how to go about it. I need to somehow determine when 4m+1 is prime, and I'm pretty sure no one has done it yet. My guess is it's just as hard as determining when any given number is prime, certainly an extraordinarily difficult if not impossible task. I'll try looking at the components of the Richey Numbers of the form 20m+5 and see if there are any useful trends.