Sumsets with restricted number of prime factors

2019 ◽  
Vol 59 (2) ◽  
pp. 251-260
Author(s):  
Bing-Ling Wu
Keyword(s):  
Restricted Number ◽  
Prime Factors

Past, Present, Future: An Overview of Roman Malta

2021 ◽  
Vol 7 (1) ◽  
pp. 231-255
Author(s):  
David Cardona
Keyword(s):  
Seventeenth Century ◽  
Predictive Model ◽  
Clear Picture ◽  
Maltese Islands ◽  
Restricted Number ◽  
The Subject ◽  
New Research

Abstract Roman Malta has been the subject of numerous historical and archaeological studies since the seventeenth century. However, the lack of documented excavations and the restricted number of sites – particularly those within the boundaries of the two main Roman towns – meant that numerous grey areas persist in our understanding of the islands under Roman rule, regardless of how many studies have been done so far. This article attempts to provide an overview of past works, studies and a discussion of the known consensus on knowledge of sites, populations and economies. This in an attempt to provide a clear picture of what we know (and what we do not) about Roman Malta. Finally, I will comment on current and new research and projects which are being carried out by various local entities and foreign institutions to enhance our knowledge of this very important historic era for the Maltese islands. This culminates into a proposal for the use of a predictive model that may help us identify new sites and, consequently, provide new data on this phase.

On the prime factors of the iterates of the Ramanujan τ–function

2020 ◽  
Vol 63 (4) ◽  
pp. 1031-1047
Author(s):  
Florian Luca ◽  
Sibusiso Mabaso ◽  
Pantelimon Stănică
Keyword(s):  
Positive Integer ◽  
Prime Factors

AbstractIn this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

scholarly journals On integers free of large prime factors

1986 ◽  
Vol 296 (1) ◽  
pp. 265-265 ◽  
Author(s):  
Adolf Hildebrand ◽  
G{érald Tenenbaum
Keyword(s):  
Prime Factors

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals Ergodic averages with prime divisor weights in

2017 ◽  
Vol 39 (4) ◽  
pp. 889-897 ◽  
Author(s):  
ZOLTÁN BUCZOLICH
Keyword(s):  
Dynamical System ◽  
Ergodic Theorem ◽  
Prime Divisor ◽  
Ergodic Averages ◽  
Weighting Functions ◽  
Pointwise Ergodic Theorem ◽  
Prime Factors ◽  
Ergodic Dynamical System ◽  
Image Position

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$, $$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$ This answers a question raised by Cuny and Weber, who showed this result for $L^{p}$, $p>1$.

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