Some general theorems about a class of sets of numbers



Abstract

We prove a theorem which unifies some formulas, for example the counting function of some sets of numbers including all positive integers, h-free numbers, h-full numbers, etc. We also establish a conjecture and give some examples where the conjecture holds.


Keywords

sets of numbers; counting function; general theorems; least prime factor

G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford, at the Clarendon Press, 1960.

A. Ivić, The Riemann Zeta-Function, Dover Publications, Inc., Mineola, NY, 2003.

A. Ivić and P. Shiu, The distribution of powerful integers, Illinois J. Math. 26 (1982), no. 4, 576–590.

R. Jakimczuk, The kernel of powerful numbers, Int. Math. Forum 12 (2017), no. 15, 721–730.

R. Jakimczuk and M. Lalín, Sums of ω(n) and Ω(n) over the k-free parts and k-full parts of some particular sequences, Integers 22 (2022), Paper No. A113, 22 pp.

I. Niven, Averages of exponents in factoring integers, Proc. Amer. Math. Soc. 22 (1969), 356–360.

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Published : 2023-12-13


JakimczukR. (2023). Some general theorems about a class of sets of numbers. Annales Mathematicae Silesianae, 38(2), 314-335. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/16499

Rafael Jakimczuk  jakimczu@mail.unlu.edu.ar
Departamento de Ciencias Básicas, División Matemática, Universidad Nacional de Luján, Buenos Aires  Argentina



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