A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors $$p_{1},\ldots, p_{t}$$ p 1 , … , p t of n and positive integers $$\alpha_{1}, \ldots , \alpha_{t}$$ α 1 , … , α t such that $$n = p_{1}^{\alpha_{1}}+ \cdots + p_{t}^{\alpha_{t}}$$ n = p 1 α 1 + ⋯ + p t α 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 by m with t = 3 if and only if $$8 \nmid m$$ 8 ∤ 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 $${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$$ exp ( c ( log log x ) 2 / 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.

中文翻译：

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关于弱素可加数的个数

如果 n 至少有两个不同的素数因数并且存在不同的素数因数 $$p_{1},\ldots, p_{t}$$ p 1 , … , pt of n 和正整数 $$\alpha_{1}, \ldots , \alpha_{t}$$ α 1 , ... , α t 使得 $$n = p_{1}^{\alpha_{1}}+ \cdots + p_ {t}^{\alpha_{t}}$$ n = p 1 α 1 + ⋯ + pt α t 。Erdős 和 Hegyvári [2] 证明，对于任何素数 p ，都存在一个弱素数可加数，它可以被 p 整除。最近，Fang 和 Chen [ 3 ] 证明了对于任何给定的正整数 m ，当且仅当 $$8 \nmid m$$ 8 ∤ m 可以被 m 整除且 t = 3 有无穷多个弱素可加数。在本文中，我们证明对于任何给定的正整数 m ，可被 m 整除且小于 x 的弱素加性数的个数大于 $${\rm exp}(c({\rm log log} x)^{2}/ {\rm log log log} x)$$ exp ( c ( log log x ) 2 / log log log x ) 对于所有足够大的 x ，其中 c 是一个正绝对常数。常数 c 取决于算术级数中最小素数的结果。