Next primes of the form $n^n - (n-1)^{n-1} - (n-2)^{n-2} - (n-3)^{n-3} - ...- 3^3 - 2^2 - 1^1 - 0^0$

With the convention $0^0=1$, let

• $Q(1) = 1^1 – 0^0=1-1=0$
• $Q(2) = 2^2 – 1^1 – 0^0=4-1-1=2$
• $Q(3) = 3^3 – 2^2 – 1^1 – 0^0=27-4-1-1=21$
• $Q(4) = 4^4 – 3^3 – 2^2 – 1^1 – 0^0=256-27-4-1-1=223$
• and so on…

I found that $Q(2)=2$, $Q(4)=223$ , and $Q(7)=773{,}473$ are prime , but after that I didn’t find anymore primes up to $Q(1000)$. I found these some regular patterns:

• $Q(4n+1)$ and $Q(4n+2)$ will always be even numbers for all integers $n ge 0$
• $3$ will always be the LEAST PRIME FACTOR of $Q(36n+3) , Q(36n+8) , Q(36n+16) , Q(36n+19) , Q(36n+24)$ and $Q(36n+35)$, for all integers $n ge 0$

I found some semiprimes but I didn’t find any primes besides the already known $Q(2), Q(4)$, and $Q(7)$. Could you find the next prime(s) of such form ?