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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.