arXiv subjects

Giuseppe Melfi

Publications and source records attributed to Giuseppe Melfi.

5 recordsLinked to original sources

arXiv2018

Primitive weird numbers having more than three distinct prime factors

In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a given number of distinct prime factors are presented. These algorithms yield primitive weird numbers of the form $mp_1\dots p_k$ for a suitable deficient positive integer $m$ and primes $p_1,\dots,p_k$ and generalize a recent technique developed for generating primitive weird numbers of the form $2^np_1p_2$. The same techniques can be used to search for odd weird numbers, whose existence is still an open question.

math.NT↗

arXiv2018

Primitive abundant and weird numbers with many prime factors

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $Ω$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $Ω=6$, count all PANs and square-free PANs up to $Ω=7$ and count all odd PANs and odd square-free PANs up to $Ω=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example existing, the previous one being 5328 digits long [Melfi]. We find hundreds of PWNs with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWNs with at least one odd prime factor with multiplicity greater than one and $Ω= 7$ and prove that there are none with $Ω< 7$. Regarding PWNs with a cubic (or higher) odd prime factor, we prove that there are none with $Ω\le 7$, and we did not find any with larger $Ω$. Finally, we find several PWNs with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples.

math.NT↗

arXiv2005

LAD Regression and Nonparametric Methods for Detecting Outliers and Leverage Points

The detection of influential observations for the standard least squares regression model is a question that has been extensively studied. LAD regression diagnostics offers alternative approaches whose main feature is the robustness. In this paper a new approach for nonparametric detection of influencial observations in LAD regression models is presented and compared with other classical methods of diagnostics.

math.ST↗

arXiv2004

On certain positive integer sequences

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

math.NT↗

arXiv2004

On simultaneous binary expansions of $n$ and $n^2$

A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.

math.NT↗