A Folklore Divisor–Sum Obstruction for the Odd Distinct Covering Problem

A \emph{covering system} is a finite collection of residue classes whose union is all of $\Z$.
Erd\H{o}s and Selfridge asked whether there exists a \emph{distinct} covering system in which all moduli are odd.
This note records a simple necessary condition, based only on reciprocal sums of divisors of the overall modulus,
and uses it to obtain an explicit lower bound on the least possible overall modulus of a hypothetical odd distinct covering system.