On the number of weakly prime-additive numbers

Abstract

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors $p_{1},\ldots, p_{t}$ of n and positive integers $\alpha_{1}, \ldots , \alpha_{t}$ such that $n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}$. Erdős and Hegyvári [2] proved that, for any prime p, there exists a weakly prime-additive number which is divisible by p. Recently, Fang and Chen [3] proved that for any given positive integer m, there are infinitely many weakly prime-additive numbers which are divisible bym with t = 3 if and only if $8 \nmid m$. In this paper, we prove that for any given positive integer m, the number of weakly prime-additive numbers which are divisible by m and less than x is larger than ${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$ for all sufficiently large x, where c is a positive absolute constant. The constant c depends on the result on the least prime number in an arithmetic progression.