refined solution

Here is my refinement for the solution:

1. assume n=p^5. we have 1+p^5 = 5(p+p^2+p^3+p^4) = 5p(1+p+p^2+p^3) = 5p(1+p)(1+p^2).
On the otherhand, 1+p^5 = (1+p)(p^4-p^3+p^2-p+1) = (1+p)[p(p-1)(p^2+1)+1].
(1+p)[p(p-1)(p^2+1)+1] / p(1+p)(1+p^2) = p-1 + 1/[p(1+p^2)]. Obviously p-1 + 1/[p(1+p^2)] can never be 5!

2. assume n=p*q^2. we have 1+p*q^2=5(1+q)(p+q). rewrite equation as 1 = 5p + 5q + 5pq +(5-p)q^2 we have p>=7; rewrite equation as 1 = 5p +5q + 5q^2 + pq(5-q) we have q>=7.

3. 1+pq^2= (p+1) + p(q+1)(q-1) = 5(1+q)(p+q), so q+1 | (p+1) and hence p>q.
(q-1)p < (p+1)/(q+1) + (q-1)p = (1+p*q^2)/(q+1) =5(p+q) < 10p, so q < 11. we already know that q>=7. So q must be 7. We assign 7 into the original equation, luckily we have p=31.

4. The only qualified n=31*7*7.