An asymptotic formula in additive number theory

1 . Introduction . In his paper [1], Erdös introduced the sequences of positive integers b 1 < b, < . . ., with (b ;, bj ) = 1, for i ~A j, and 'bi 1 < oo . With any such arbitrary sequence of integers, he associated the sequence {di} of all positive integers not divisible by any bj , and he showed that if b1 > 2, there exists a 0 < 1 (independent of the sequence {b i }) such that d i 1 di < d°, for i > i o . Later, Szemerédi [4] made an important progress on the problem, showing that 0 can be taken to be any number greater than In this paper, we study this sequence from a different point of view . We study the number N (n) of solutions of the equation n = p + d, where p is a prime and d 4' 0(modbj ) for any j . In fact we derive an asymptotic formula for N(n), when b 1 > 3. We also study N(n) when the condition (b i , bj ) = 1 is dropped .