Let σ(n) be the sum of the positive divisors of an integer n and ϕ(n) be the number of positive integers smaller than n that are coprime to n. If p is a prime number, what is the maximum value σ(p)ϕ(p)?

Answer 1

The actual value can be found at

http://www.proofwiki.org/wiki/Product_of_Sigma_and_Euler_Phi_Functions

Since the value is n^2 times a bunch of factors each less than 1, I guess n^2 is the maximum possible value.