# On the number of weakly prime-additive numbers

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## 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 by
*m* 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.

## Key words and phrases

prime-additive number Chinese remainder theorem Dirichlet’s theorem representation of integers## Mathematics Subject Classification

11A07 11A41## Preview

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## Notes

### Acknowledgement

We sincerely thank the referee for valuable suggestions.

## References

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