Problem 470 — Visual explainer (undergrad)
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Problem statement
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Call $n$ weird if $\sigma(n)\geq 2n$ and $n$ is not pseudoperfect, that is, it is not the sum of any set of its divisors.

Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?

Picture
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Divisors / subset-sum picture:

divisors(n):  1, d1, d2, ..., n
choose some of them and add them up

σ(n) = sum of *all* divisors

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: multiplicative number theory.

Benchmarks / known results (from comments)
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- Weird numbers were investigated by Benkoski and Erd\H{o}s, who proved that
  the set of weird numbers has positive density.
- The smallest weird number is $70$.
- Melfi has proved that there are infinitely many primitive weird numbers,
  conditional on the fact that $p_{n+1}-p_n<\frac{1}{10}p_n^{1/2}$ for all
  large $n$, which in turn would follow from well-known conjectures concerning
  prime gaps.
- Fang has shown there are no odd weird numbers below $10^{21}$, and Liddy and
  Riedl have shown that an odd weird number must have at least 6 prime
  divisors.