Sum of GCDs

Number Theory Level 3

A positive integer $n$ is called ingenious if $\displaystyle\sum_{m=1}^{n} \gcd (m,n)$ is a prime. Find the number of ingenious integers between $3$ and $100$ (inclusive).

Details and assumptions

You may refer to this list of primes.
As an explicit example, when $n=3,$ we have $\displaystyle\sum_{m=1}^{3} \gcd (m, 3) = \gcd (1, 3) + \gcd (2, 3) + \gcd (3, 3)= 5,$ which is a prime.
This problem was inspired by IMOSL 2004 N2.