Following a statement of the well-known Erdos-Turán conjecture, Erdos mentioned the following even stronger conjecture: if the n-th term a_n of a sequence A of positive integers is bounded by , for some positive real constant , then the number of representations of n as a sum of two terms from A is an unbounded function of n. Here we show that if a_n differs from (or from a quadratic polynomial with rational coefficients q(n)) by at most , then the number of representations function is indeed unbounded.