Prime factorization of numbers in $1+6N$ and $5+6N$?

Question

In my thesis I need to prove a certain statement for the cases where p is in 1+6N or in 5+6N.

For example $5+6N$: I checked the numbers until $200$ and it seems as if all numbers in this set are - prime - the product of two distinct numbers - a power of a certain prime number

If I could show that this this is true for all numbers in $5+6N$, then I could finish my proof. Do you have any ideas or are there some properties of numbers out of this set? Some properties about the prime factorization of $p$ in $5+6N$ and $1+5N$ would be helpful..

Answer 1

$$ 7 \cdot 13 \cdot 19 \equiv 1 \pmod 6 $$ $$ 5 \cdot 11 \cdot 17 \equiv 5 \pmod 6 $$ for example

$$ 1729 = 1 + 6 \cdot 288 $$ $$ 935 = 5 + 6 \cdot 155 $$