Problem #825: Is there an absolute constant $C>0$ such that every integer...

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Progress

A problem of Benkoski and Erdős. In other words, this problem asks for an upper bound for the abundancy index of weird numbers. This could be true with $C=3$. We must have $C>2$ since $\sigma(70)=144$ but $70$ is not the distinct sum of integers from $\{1,2,5,7,10,14,35\}$.

Erdős suggested that as $C\to \infty$ only divisors at most $\epsilon n$ need to be used, where $\epsilon \to 0$.

Weisenberg has observed that if $n$ is a weird number with an abundancy index $\geq 4$ then it is divisible by an odd weird number. In particular, if there are no odd weird numbers (see [470]) then every weird number has abundancy index $<4$. Indeed, if $l(n)$ is the abundancy index and $n=2^km$ with $m$ odd then $l(n)=l(2^k)l(m)$, and $l(2^k)<2$ so if $l(n)\geq 4$ then $l(m)>2$, and hence $m$ is weird (as a factor of a weird number).

A similar argument shows that either there are infinitely many primitive weird numbers or there is an upper bound for the abundancy index of all weird numbers.

See also [18] and [470].

This is part of problem B2 in Guy's collection [Gu04] (the \$25 is reported by Guy as offered by Erdős for a solution to this question).