**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 $n$.
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I will survey some known results about weird numbers.
For example there are infinitely many weird numbers and indeed
their asymptotic density is positive.
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I will
speak about some old questions about
weird numbers raised by Benkoski and Erd\H os in the '70s.
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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.
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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.
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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.

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

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