In Theorem 1 a set of sufficient conditions is investigated for the su
mmability factor of type theta(n) is an element of (\(N) over bar, q\k
, \(N) over bar, p\k), k greater than or equal to 1. In the special ca
se when k = 1 three different sets of necessary and sufficient conditi
ons are established in Theorem 2 for the factor theta(n). Various know
n and some new results are also deduced as special cases.