An equation of fractional parts

Find the sum of all positive integers \(n\) such that for some positive integer \( m\geq n\), the following inequality is satisfied for all real \(x\): \[ \left\{ \frac{x}{15} \right\}- \left\{ \frac{x}{n} \right\}-\left\{ \frac{x}{m} \right\} \leq 0.\]

Details and assumptions

The function \(\{a\}\) denotes the fractional part of \(a\). It can be calculated as \( \{ a\} =a-\lfloor a \rfloor\). As an explicit example, \(\{5\}=0\), \(\{3.75\}=0.75\), \(\{-4.2\}=0.8\) .

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