Even perfect numbers and their Euler's function (Q1089024)

A positive integer n is called perfect if \(\sigma (n)=2n\), where \(\sigma\) (n) denotes the sum of all positive divisors of n. While the problem of the existence of odd perfect numbers is still open, a well-known theorem of Euclid and Euler says that an even positive integer n is perfect if and only if there exists a Mersenne prime \(2^ p-1\) such that \(n=2^{p- 1}(2^ p-1)\). Starting from this classical result the author draws some straightforward conclusions about even perfect numbers such as ''If \(n=2^{p-1}(2^ p-1)\) is an even perfect number, then \(\phi (n)=n- 4^{p-1}\)''. Here as usual \(\phi\) denotes Euler's totient function.