It's related to Dottie number and prime factorization .
Let :
$D=\operatorname{Dottie-number}\simeq 0.7390851332$
Now define $n\geq 3$ an integer:
$$\lfloor\left(D\right)^{-2n}\rfloor=P$$
Then it seems that the prime factorization is write as for example :
$$97636=2^2×7×11×317,75566970=2×3^2×5×839633$$
So I conjecture that the prime factorization of $P$ is composed with one prime with a power less than it self and other prime with power equal to zero or one .
It have some similarity with another question of mine with $n!+1$
Added after lulu's answer :
Can you give a counter-example if the last digit o $P$ is not even i.e $(0,2,4,6,8)$ ?
Some example at $n=29$ to $n=50$
1)$n=29$ last digit $8$ exclude
2)$n=30$ last digit is $0$ exclude
3)$n=31$ $138338505 = 3^2×5×3074189$ OK
4)$n=32$ $253252735 = 5×50650547$ OK
5)$n=33$ $463623255 = 3^2×5×71×145109$ OK
6)$n=34$ the last digit is $4$ exclude
7)$n=35$ the last digit is a $8$ exclude
8)$n=36$ $2844450715 = 5×421×1351283$ OK
9)$n=37$ the last digit is zero $exclude$
10)$n=38$ $9532801445 = 5×877×2173957$ OK
11)$n=39$ $17451453903 = 3×13×6637×67421$ OK
12)$n=40$ $31947926857 = 7×11×3461×119881$ OK
13)$n=41$ $58486246255 = 5×31×19423×19427$ OK
14)$n=42$ $107069263563 = 3×35689754521$ OK
15)$n=43$ last digit is a $4$ exclude
16)$n=44$ $358828516513 = 211×12967×131149$ OK
17)$n=45$ last digit is a $2$ exclude
18)$n=46$ $1202566450709 = 13×199×4663×99689$ OK
19)$n=47$ $2201507405719 = 163741×13445059$ OK
20)$n=48$ $4030242864815 = 5×17×409×991×116981$
21)$n=49$ $7378061734971 = 3^2×819784637219$ near a counter-example but OK
22)$n=50$ $13506827452083 = 3^2×7×21221×10102921$ near a counter-exmple but OK
Question :
Can you (dis)prove the conjecture ?
Reference :
For the benefit of those unfamiliar with the term, "the Dottie Number" refers to the real solution of $\cos x = x$.
The numerical remark seems like just an accident of small numbers. Many instances occur where $P_n=\lfloor D^{-2n}\rfloor$ is divisible by $2^2$ or $3^3$. And $$P_{28}=\big \lfloor D^{-56}\big \rfloor =22548096 = 2^7\times 3^2\times 23^2\times 37$$
see wolfram alpha