Third Italian Number Theory Meeting

# On some questions about weird numbers

speaker: Giuseppe Melfi (University of Neuchâtel)

abstract: A weird number is a positive integer $n$ whose sum of its proper divisors exceeds $n$ but no sums of distinct proper divisors equals $n$. \\ I will survey some known results about weird numbers. For example there are infinitely many weird numbers and indeed their asymptotic density is positive. \\ I will speak about some old questions about weird numbers raised by Benkoski and Erd\H os in the '70s. \\ One of the most intriguing open questions is the existence of odd weird numbers. All known weird numbers are even, and recent computations only showed that there are no odd weird numbers up to $10{21}$. Erd\H os offered 10\$for the first example of an odd weird number and 25\$ for the proof that no odd weird number exists. \\ If $w$ is a weird number and $p>\sigma(w)$, then $wp$ is a weird number. As a consequence of this elementary property, \textsl{primitive} weird numbers are defined as those weird numbers that are not a multiple of another weird number. \\ In a recent paper I showed that under the assumption of the Cram\'er conjecture, or even of much weaker conjectural assumptions on the distribution of primes in short intervals, there exist infinitely primitive weird numbers, another problem raised by Erdos and Benkoski.

timetable:
Tue 22 Sep, 15:00 - 15:25, Aula Russo
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