On an additive representation function

Let A be an infinite subset of natural numbers, and X a positive real number. Let r(n) denotes the number of solution of the equation n=a1+a2 where a1 a2 and a1, a2 A. Also let A(X) denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)≠1, we improve the lower bound of A(X) given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible.