In my first attempt at the problem it made this mistake: "The condition fails for i=1 since 1 does not divide p+q unless p+q is a multiple of n, which is not g

In my first attempt at the problem it made this mistake: "The condition fails for i=1 since 1 does not divide p+q unless p+q is a multiple of n, which is not generally true" (1 divides everything) In the second it gives up: "However, this conclusion is based on heuristic reasoning and examples" Third attempt it makes this mistake: "If the immediate next divisor, di+1 , is not a multiple of p (for instance, it could be q or a product involving q), then p does not divide di+1. Hence, p will not divide the sum di+1+di+2 in such a case, violating the condition." (the fact that p does not divide d_i+1 does not imply that it does not divide d_i+1, d_i+2) Then it gives up again: "However, this conclusion is based on heuristic reasoning and examples" Then it makes this basic logic mistake: "However, since p and q are distinct primes, p does not divide q, and it's not guaranteed that p divides q+d_i+2 , especially if d_i+2 is not a multiple of p. Therefore, for such n, the condition fails." (the fact that "it's not guaranteed" doesn't imply that it's false) Overall it proves that prime powers have the property, conjures that non-prime-powers don't, proves that p*q doesn't have it, but completely fails at coming up with a proof strategy that proves that non-prime-powers don't have the property.