I don't know if this concept is already defined, I consider the number $n$ to be a perfect prime (weak) if $n$ is a prime, its digits are primes and the sum of the digits is prime. Every one digit prime is a perfect prime (weak) and it's clear that every perfect prime has to be a combination of elements in the set $ \{2,3,5,7 \}$. Some examples of these are 23,223,227,537233,777277. So my question is, there always exist infinite $(d_1,d_2,d_3,d_4) \in \mathbb{Z}^4 $ such that $$ 2d_1 +3d_2 + 5d_3 + 7d_4 $$ is prime and some combination of the $d_i$ $2,3,5,7$ is prime?
The perfect primes (strong) or perfect primes complete the "perfect", as the product of the digits can not be prime we put the condition that the product is a perfect number $(6,28,496,...)$ For example, 23 is a perfect prime (weak) and $2*3=6$ a perfect number, so $23$ is a perfect prime. $227$ is a perfect prime (weak) and $2*2*7=28$ a perfect number, so $227$ is a perfect prime. The formula $2^{p-1}(2^p-1) $ with $p$ and $2^p-1$ prime generates perfect numbers, but it is not known that every perfect number is the form $2^{p-1}(2^p-1) $. No odd perfect number is known, it's still an open problem. If every perfect number is even of the form $2^{p-1}(2^p-1) $ as before then the only perfect primes are $23$ and $227$ because the next perfect number after $28$ would be of the form $2^aP$ with $P$ a prime larger than $7$, so $P$ must have $2$ or more digits, then we can not make a number with digits $ \{2,3,5,7 \}$ sucht the product is $2^aP$