I was reading the following theorem.
Theorem. There are infinitely-many even abundant numbers.
Proof. For a positive integer $a$, let $n=2^a\cdot3$, and compute$\sigma(2^a\cdot3)=\sigma(2^a)\sigma(3)=(2^{a+1}-1)(3+1)=2(2^a\cdot3)+2^{a+1}-4=2n+2^{a+1}-4,$ which is greater than $2n$ whenever $a\geq2$. This demonstrates infinitely-many even abundant numbers. $\square$
I know that this proof assumes knowledge of the fact that the smallest even abundant number is $12=2^2\cdot3$, and that any multiple of an abundant number is also abundant. However, It is not immediately clear to me that $2^{a+1}-4$ is a multiple of $2n$ for $n\geq2$. What am I missing?
Thanks in advance.