Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.8

Explicit Estimates Involving the Primorial Integers and Applications

Safia Aoudjit and Djamel Berkane
University of Blida
Department of Mathematics
P. O. Box 270, Route de Soumaa

Pierre Dusart
Faculté des sciences et techniques
Université de Limoges
P. O. Box 123, avenue Albert Thomas
87060 Limoges Cedex


In this paper, we propose several explicit bounds for the function that counts the number of primorial integers less than or equal to a given positive real number. As applications, we obtain an effective version of Pósa's inequality, and a method to estimate the maximal value of the sum over prime divisors Σp|q f(p) for a positive decreasing function f, when q ranges over all integers less than x. In particular, we improve the log p upper bound for the maximal value of Σp|q (log p)/(p—1).

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(Concerned with sequences A000040 A002110 A056127 A111972 A233824.)

Received February 6 2021; revised versions received February 17 2021; February 19 2021; July 11 2021; July 28 2021. Published in Journal of Integer Sequences, July 28 2021.

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