We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a
-automatic sequence is itself
-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length
-automatic sequence is
-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.