We consider the class P-1 of all infinite words x is an element of A(omega) over a finite alphabet A admitting a prefixal factorization, i.e., a factorization x = U0U1U2...where each U-1 is a non-empty prefix of x. With each x is an element of P-1 one naturally associates a "derived" infinite word delta(x) which may or may not admit a prefixal factorization. We are interested in the class P-infinity of all words x of P-1 such that delta(n)(x) is an element of P-1 for all n >= 1. Our primary motivation for studying the class P-infinity stems from its connection to a coloring problem on infinite words independently posed by T. Brown and by the second author. More precisely, let P be the class of all words x is an element of A(omega) such that for every finite coloring phi : A(+) -> C there exist c is an element of C and a factorization x = V0V1V2 ...with phi(V-i) = c for each i >= 0. In a recent paper (de Luca et al., 2014), we conjectured that a word x is an element of P if and only if x is purely periodic. In this paper we prove that P not subset of P-infinity, so in other words, potential candidates to a counter-example to our conjecture are amongst the non-periodic elements of P-infinity. We establish several results on the class P-infinity. In particular, we prove that a Sturmian word x belongs to,P-infinity, if and only if x is nonsingular, i.e., no proper suffix of x is a standard Sturmian word. (C) 2015 Elsevier Ltd. All rights reserved.