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Explicit Estimates Involving the Primorial Integers and Applications
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Safia Aoudjit and Djamel Berkane

University of Blida

Department of Mathematics

P. O. Box 270, Route de Soumaa

Blida

Algeria

Pierre Dusart

XLIM UMR 7252

Faculté des sciences et techniques

Université de Limoges

P. O. Box 123, avenue Albert Thomas

87060 Limoges Cedex

France

**Abstract:**

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
Σ_{p|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
Σ_{p|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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