1610.31 – Nasty Numbers

Let us say that a positive integer is $nasty$ if it has two different factorizations such that the difference between the two factors of one of the factorizations equals the sum of the two factors of the other factorization. For example, $6$ is nasty because $2 + 3 = 6 - 1$ and 24 is nasty because $4 + 6 = 12 - 2$. And 30 is nasty because $3 + 10 = 15 - 2.$

Find all nasty numbers less than 180. There are 6 besides the examples. (This is a nifty problem to do on a computer.)