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Let $\dfrac{\sigma(f_0)+h}{f_0}=\dfrac{\sigma(f_1)}{f_1}=\dfrac{\sigma(f_2)}{f_2}=\cdots=\dfrac{\sigma(f_s)}{f_s}$, where $\gcd(\sigma(f_0)+h,f_0)=1$, $h>0$, and $f_0<f_1<\cdots<f_s$. Let $r_i$ be the least positive integer such that every $1\leq m\leq(r_i-1)\sigma(f_i)$ has a representation as a sum of at most $r_i-1$ of each of the divisors of $f_i$. Conjecture: $r_0\geq r_1\geq\cdots\geq r_s$.

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