For what values of n is $2^{2n} -1$ divisible by $4n+1$

For what values of n is the expresion $2^{2n} -1$ is divisible by $4n+1$.

I have checked using a computer and the values of 2n I get are 8,20,36,44,48,56,68,96,116,120,128,140,156,168,170,176 $\cdots$etc [For $2n<200$] .

I don't seem to find a relation between these values.I am curious to see if we can find a relation by applying simple number theory.

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