It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. We know that $\phi(p^k) = p^{k-1} (p-1)$ for all primes $p$.

What about the multiplicative function $\psi$ defined on the primes by $\psi(p^k) = p^{k-1} (p+1)$? Does it have a name? If so, what's known about it?

For example, can one evaluate $\sum_{n \leq X} \psi(n)$?