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.