Primes of Atlantis: a definition and a problem

Question

We denote the floor function as $\lfloor x\rfloor$, and for integers $k\geq 1$ we consider the following sum of the areas of three consecutive circles $$\pi k^2+\pi(k+1)^2+\pi(k+2)^2=\pi \left( 3k^2+6k+5\right). $$

Definition. When for an integer $n\geq 1$ the integer $$\mathcal{A}(n)=\lfloor \pi\left( 3n^2+6n+5\right) \rfloor$$ is a prime number, I say that it's a prime of Atlantis.

Our sequence of prime of Atlantis starts as $$43, 157, 241, 769, 4567, 11551, 14341, 16631, 19949\ldots$$ corresponding to the indexes $n's:1, 3, 4, 8, 21, 34, 38, 41, 45, \ldots$ as you can see with these codes using Wolfram Alpha online calculator:

Table IsPrime(floor(pi (n^2+(n+1)^2+(n+2)^2))), for n=1 to 100

Table floor(pi (n^2+(n+1)^2+(n+2)^2)), for n=1 to 100

Question. I would like to know if we can deduce if there are infinitely many primes of Atlantis. Many thanks.

If you can't solve the problem, but you can provide us useful reasonings or calculations about the Question, please share your knowledges.

Answer 1

This is not a complete answer , but I show some statistics about the "primes of Atlantis"

Program PARI/GP is used

First of all the $n$ upto $1000$, for which we get a "prime of Atlantis" :

\p 10000 { High precision calculation }
pa(n)=isprime(truncate(Pi*(3*n^2+6*n+5)),2)==1 {pa(n) is true if and only if we get a prime of Atlantis by choosing $n$}

? select(m->pa(m)==1,[0..1000])
%4 = [1, 3, 4, 8, 21, 34, 38, 41, 45, 69, 75, 91, 136, 166, 179, 190, 202, 222,
227, 228, 229, 230, 239, 267, 284, 308, 313, 317, 323, 351, 359, 381, 392, 409,
417, 429, 433, 434, 442, 449, 456, 460, 463, 468, 486, 490, 518, 527, 549, 585,
588, 607, 632, 651, 668, 670, 684, 694, 700, 703, 727, 764, 775, 782, 805, 811,
814, 820, 821, 844, 850, 894, 896, 920, 925, 926, 932, 969, 979, 985]
?

For $n\in [1,10]$ , we get $4$ primes

For $n\in [10,100]$ , we get $8$ primes

For $n\in [100,1000]$ , we get $68$ primes

For $n\in [10^3,10^4]$ , we get $498$ primes

For $n\in [10^4,10^5]$ , we get $3778$ primes

For $n\in [10^5,10^6]$ , we get $31767$ primes

Some large $n$, for which we get a "prime of Atlantis"

$$10^{100}+142$$ $$10^{200}+114$$ $$10^{500}+6391$$ $$10^{1000}+1395$$

with $201$ , $401$ , $1001$ and $2001$ digits respectively. There are probably infinite many "primes of Atlantis", but it is hard to imagine that this can be proven.