Bob's nested sum

For positive integers \(m\) and \(n\), let

\[ f(m,n) = \displaystyle \sum_{a_1=1}^{m} \left(\displaystyle \sum_{a_2=1}^{a_1} \left(\displaystyle \sum_{a_3=1}^{a_2} \left(\cdots \left(\displaystyle \sum_{a_{n-1}=1}^{a_{n-2}} \left(\displaystyle \sum_{a_n=1}^{a_{n-1}} a_n\right)\right) \cdots \right)\right)\right).\]

How many values of \(n\leq 1000\) satisfy \(f(11,n) \mid f(12,n)\)?

This problem is posed by Bob K.

Details and assumptions

For \(n=1\), \( f(m,n) = \sum_{a_1=1}^m a_1 \).

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