Two Questions About Ljujié And Nathanson

Let N be the set of all nonnegative integers. For A,M C N and n ? N, let p(n,A,M)denote the number of representations of n in the form n = ?a?Amaa,where ma ? M ? {0} for all a ?A. and ma ?M for only finitely many a .By an explicit construction, Li-Xia Dai and Yong-Gao Chen proved that,for A = {n!}n?1 or for A = {nn}n?1, there exists an explicit infinite set M of positive integers so that for all n ? 1.In this note, we optimize the set M.It is proved that for A = {n!}n?1 or for A = {nn}n?1, there exist an integer m0 ? N and an explicit infinite set M, so that m ? M is the prime number for all m> m0 and for all n > 1.