proof of the correspondence between even 2-superperfect numbers and Mersenne primes

Statement. Among the even numbers, only powers of two $2^x$ (with $x$ being a nonnegative integer) can be 2-superperfect numbers (http://planetmath.org/SuperperfectNumber), and then if and only if $2^{x+1}-1$ is a Mersenne prime. (The default multiplier $m=2$ is tacitly assumed from this point forward).

Proof. The only divisors of $n=2^x$ are smaller powers of 2 and itself, $1,2,\mathrm{\dots },2^{x-1},2^x$. Therefore, the first iteration of the sum of divisors function is

$$\sigma (n)=\sum _{i=0}^x{2}^i=2^{x+1}-1=2n-1.$$

If $2n-1$ is prime, that means its only other divisor is 1, and thus for the second iteration $\sigma (2n-1)=2n$, and is thus a 2-superperfect number. But if $2n-1$ is composite then it is clear that $\sigma (2n-1)>2n$ by at least 2. So, for example, $\sigma (8)=15$ and ${\sigma }^2(8)=24$, so 8 is not 2-superperfect. One more example: $\sigma (16)=31$ and since 31 is prime, ${\sigma }^2(16)=32$.

Now it only remains to prove that no other even number $n$ can be 2-superperfect. Any other even number can of course still be divisible by one or more powers of two, but it also must be divisible by some odd prime $p>2$. Since the sum of divisors function is a multiplicative function, it follows that if $n=2^{x}p$ then $\sigma (n)=\sigma (2^x)\sigma (p)$. So, if, say, $p=3$, it is clear that $(2^{x+3}-4)>2^{x+1}3$, and that on the second iteration this value that already exceeded twice the original value will be even greater. For example, $12=2^{2}3$, and $\sigma (12)=2^5-4$ which is greater than $2^{3}3$ by 4. With any larger $p$ the excess will be much greater.

Title	proof of the correspondence between even 2-superperfect numbers and Mersenne primes
Canonical name	ProofOfTheCorrespondenceBetweenEven2superperfectNumbersAndMersennePrimes
Date of creation	2013-03-22 17:03:48
Last modified on	2013-03-22 17:03:48
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Proof
Classification	msc 11A25