On limits and formulae where functions of slow increase appear

Rafael Jakimczuk
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 4, Pages 42–51
Full paper (PDF, 168 Kb)

Details

Authors and affiliations

Rafael Jakimczuk
División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina

Abstract

Let A_n be a strictly increasing sequence of positive integers such that A_n ∼ n^sf(n), where f(x) is a function of slow increase and s is a positive real number. In this article we obtain some limits and asymptotic formulae where appear functions of slow increase. As example, we apply the obtained results to the sequence of numbers with exactly k prime factors in their prime factorization, in particular to the sequence of prime numbers (k = 1).

Keywords

• Functions of slow increase
• Integer sequences
• The e number
• Limits
• Numbers with a fixed number of primes in their prime factorization

AMS Classification

• 11A99
• 11B99

References

1. Jakimczuk, R. (2007) The ratio between the average factor in a product and the last factor, Mathematical Sciences: Quarterly Journal, 1, 53–62.
2. Jakimczuk, R. (2010) Functions of slow increase and integer sequences, Journal of Integer Sequences, 13, Article 10.1.1.
3. Jakimczuk, R. (2016) On a limit where appear prime numbers. Notes on Number Theory and Discrete Mathematics, 22(1), 1–4.
4. Sándor, J. (2012) On certain bounds and limits for prime numbers. Notes on Number Theory and Discrete Mathematics, 18(1), 1–5.
5. Sándor, J. & Verroken A. (2011) On a limit involving the product of prime numbers. Notes on Number Theory and Discrete Mathematics, 17(2), 1–3.