Rafael Jakimczuk

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 19, 2013, Number 1, Pages 55—58

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## Details

### Authors and affiliations

Rafael Jakimczuk

*División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina
*

### Abstract

Let *P*(*n*) be the greatest prime factor of a positive integer *n* ≥ 2. Let *L* (*n*) be the number of 2 ≤ *k* ≤ *n* such that *P*(*k*) > *k ^{α}*, where 0 <

*α*< 1. We prove the following asymptotic formula

where

*ρ*(

*α*) is the Dickman’s function.

### Keywords

- Greatest prime factor
- Distribution

### AMS Classification

- 11A99
- 11B99

### References

- Kemeny, J. Largest prime factor, J. Pure Appl. Algebra, Vol. 89, 1993, 181–186.
- LeVeque, W. J. Topics in Number Theory, Addison-Wesley, 1958.
- Ramaswami, R. On the number of positive integers less than
*x*and free of prime divisors greater than*x*, Bull. Amer. Math. Soc., Vol. 55, 1949, 1122–1127.^{c}

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## Cite this paper

APAJakimczuk, R. (2013). A note on the density of the Greatest Prime Factor. Notes on Number Theory and Discrete Mathematics, 19(1), 55-58.

ChicagoJakimczuk, Rafael. “A Note on the Density of the Greatest Prime Factor.” Notes on Number Theory and Discrete Mathematics 19, no. 1 (2013): 55-58.

MLAJakimczuk, Rafael. “A Note on the Density of the Greatest Prime Factor.” Notes on Number Theory and Discrete Mathematics 19.1 (2013): 55-58. Print.