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STRONGLY
$q$-ADDITIVE FUNCTIONS AND DISTRIBUTIONAL PROPERTIES OF THE LARGEST PRIME FACTOR
Published online by Cambridge University Press: 17 November 2015
Abstract
Let $P(n)$ denote the largest prime factor of an integer
$n\geq 2$. In this paper, we study the distribution of the sequence
$\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer
$b\geq 2$, where
$f$ is a strongly
$q$-additive integer-valued function (that is,
$f(aq^{j}+b)=f(a)+f(b),$ with
$(a,b,j)\in \mathbb{N}^{3}$,
$0\leq b<q^{j}$). We also show that the sequence
$\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if
${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.
MSC classification
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- Research Article
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- Copyright
- © 2015 Australian Mathematical Publishing Association Inc.
References
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