On a problem on restricted k-colored partitions

Let c(k,j) (n) be the number of (k, j)-colored partitions of n. Recently, Keith proved that for j is an element of {4, 7}, if c(9,j) (3(l) n + 2) 0 (mod 27) for all n >= 0, then l is large. We prove that such l do not exist. Furthermore, for any positive integers a, b with (a, b) = 1, there exist infinitely many positive integers n such that C-9,C-j(an+b) not equivalent to 0 (mod 27), where j is an element of {4, 7}.