  We study the (variable-length) codes of the form X u {a^n}, where X c a*wa*
and |X| = n. We extend various notions and results from factorizations of
cyclic groups theory to this type of codes. In particular, when n is the
product of at most three primes or has the form pq^k (with p and q prime), we
prove that they are composed of prefix and suffix codes. We provide
counterexamples for other n. It implies that the long-standing triangle
conjecture is true for this type of n. We also prove a conjecture about the
size of a potential counterexample to the conjecture.
