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Very Stupid Numbers

Description

For some integer \(n\), let \(D_{n_1}=n\)  and for \(i>1\), let \(D_{n_i}\) be the sum of the factors of \(D_{n_{i-1}}\).

A number \(n\) is stupid if \(D_{n_1}\) and \(D_{n_2}\) are relatively prime.  For example, \(4\) contains the factors, \(1,\,2,\) and \(4,\) which add up to \(7.\) Thus, \(D_{4_1} = 4\) and \(D_{4_2} = 7\).  Since \(\gcd(4,\,7)=1\), it follows that \(4\) is stupid.

A number \(n\) is very stupid if for every pair \(x,\,y\) in \(\{D_{n_1},\,D_{n_2},\,D_{n_3}\}\)\(x\) and \(y\) are relatively prime. For example, \(D_{7_1} = 7,\,D_{7_2}=8,\) and \(D_{7_3}=15\), and so \(7\) is very stupid.

Similarly, a number \(n\) is very very stupid if for every pair  \(x,\,y\) in \(\{D_{n_1},\,D_{n_2},\,D_{n_3},\,D_{n_4}\}\)\(x\) and \(y\) are relatively prime.


Question

Find the sum of all very very stupid numbers between \(1\) and \(1000\), inclusive.


Find the sum of all very very stupid numbers between \(1\) and \(1000\), inclusive.