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On the number of weakly prime-additive numbers
- Source :
- Acta Mathematica Hungarica. 160:444-452
- Publication Year :
- 2019
- Publisher :
- Springer Science and Business Media LLC, 2019.
-
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 Hegyvari [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.
- Subjects :
- Mathematics::Number Theory
General Mathematics
010102 general mathematics
Prime number
010103 numerical & computational mathematics
01 natural sciences
Prime (order theory)
Combinatorics
Integer
Log-log plot
Arithmetic progression
0101 mathematics
Absolute constant
Constant (mathematics)
Mathematics
Subjects
Details
- ISSN :
- 15882632 and 02365294
- Volume :
- 160
- Database :
- OpenAIRE
- Journal :
- Acta Mathematica Hungarica
- Accession number :
- edsair.doi...........09c95360095cdb509cfc3e76f418ce76