Primate

Since Primate number is composite , it has more than $2$ positive divisors , and therefore has (prime) odd number of positive divisors
Remembering that for natural $n > 1$ , $n = p_1^{q_1}p_2^{q_2} \cdots p_k^{q_k}$ , where $p_1 , p_2 \cdots p_k$ are different primes , then , $n$ has $(q_1 + 1)(q_2 + 1) \cdots (q_k + 1)$ positive divisors . So , in order to get (odd) prime number , we must have $n$ is a (even) power of a prime number .
Considering the possible unit digit of squared number (i.e. $1 , 4 , 5 , 6 , 9 , 0)$ and notice that Primate number must have all of its digits prime , therefore , Primate number is an even power of a prime with unit digit $5$ . The only such prime is $5$ . Therefore , all Primate number have form $5^{2a} = 25^a$ , where $2a + 1$ is prime .
Notice that for $a > 1$ , the last three digit of $25^a$ is $625$ , and $6$ is not a prime .
Therefore , the only Primate number is $25$ . And , the sum of its digits is $7$ , which is a prime number . Therefore , all Primate number have prime digital sum .