The following questions and close problems are studied. (i) Is it true that T is p-nuclear provided that T** is p-nuclear? (ii) Is it true that T is dually p-nuclear provided that T* is p-nuclear? (iii) Is it true that if T* is compactly factorable in the space l_p, then T is (strictly) factorable in the space l_p′? Here, T* is the adjoint operator of a bounded operator T : X → Y in Banach spaces X and Y. Bibliography. 30 titles.