The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below).
Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \to \mathbb{N}_{SF}$ defined by $$ f(n)=\prod_{\substack{p \text{ prime, with} \\ p|n \text{ or } (p-1)|n}} p.$$ Let us iterate this map and consider the sequences of the form $(f^{(r)}(n))_r$. Note that $n$ divides $f(n)$, so this sequence is weakly increasing. If it is strictly increasing then $n$ is called explosive (otherwise the sequence is eventually constant).
Example: $(f^{(r)}(1))_r = (1, \ 2, \ 2^1 3, \ 2^1 3^1 7, \ 2^1 3^1 7^1 43, \ 2^1 3^1 7^1 43, \dots)$ is eventually constant.
Let $u_r(n)$ be the numbers of prime factors of $f^{(r)}(n)$, with $r \ge 0$. Then for example:
- $(u_r(1))_r = (0,1,2,3,4,4,\dots)$,
- $(u_r(5))_r = (1, 2, 4, 9, 77, \dots)$.
Observe that the numbers of divisors of $f^{(4)}(5)$ is $2^{77} > 10^{23} $, so $u_5(5)$ should be huge, while its computation is unreachable directly. Then $5$ looks explosive, but a proof seems unreachable too.
What may be reachable is the proof (by contradiction) of the existence of an explosive prime.
Question 1: Is there an explosive prime?
It should be useful to recall that the density of primes of the form $m+1$ with $m$ square-free (in the set of all primes) is about $1/3$, more precisely Artin's constant (see why here):
$$ c_{Artin}:= \prod_{p \ prime} (1-\frac{1}{(p-1)p}) = 0.3739558136192\dots $$
It is good to mention that the statement "$13$ and $2287$ are explosive" implies Carmichael's totient conjecture, see why in this paper by Kevin Ford (on page 38): $13$ goes to his case I (as $13 = 12 \times 1 + 1$) and $2287$ to his case II (as $2287 = 18 \times 127 +1$ and $127 = 18 \times 7 + 1$). But above statement looks true experimentally:
$(u_r(13))_r = (1, 2, 3, 5, 9, 24,\dots)$,
$(u_r(2287))_r = (1, 2, 3, 5, 7, 8, 11, 24,\dots)$.
I checked that $u_6(13)>10000$, whereas $2^{10000}>10^{3000}$, so $u_7(13)$ should be really huge.
Finally, about the density of explosive primes in the set of all primes: among the $168$ primes less than $1000$, $119$ ones looks explosive (as $13$), i.e. more than $70 \%$. Here are the $49$ other ones (which are non-explosive, confirmed): $2, 3, 7, 19, 43, 67, 79, 97, 127, 163, 193, 211, 223, 307, 317, 337, 349, 409, 421, 463, 487, 499, 521, 523, 541, 547, 569, 571, 617, 631, 643, 673, 691, 709, 733, 739, 757, 787, 809, 811, 823, 839, 853, 857, 859, 883, 919, 967, 997.$
Assuming that Question 1 has a positive answer:
Question 2: What is the density of explosive primes in the set of all primes?
My guess is that the set of explosive primes is infinite, but of density zero in the set of all primes, and also that a proof of these two statements is reachable.
Experimentally, there is $c>0$ such that the number of looking explosive primes less than $n$ is about $c\frac{n}{\log(n)^2}$ for $n$ large enough, as for the Sophie Germain primes or the twin primes, but with a different constant $c$ (apparently): $c=1$ for the explosive ones, $c \sim 1.32032$ (more precisely, two times the Hardy–Littlewood's twin prime constant) for the other ones. The following table provides the counting of such primes between $10^n$ and $10^n + 10^6$ for $n=23,24,25$, together with the prediction of the estimate for $c=1$ and $c=1.32032$: $$\begin{array}{c|c|c|c|c|c} n & \# \text{ Explosive} & c=1 & \# \text{ Sophie Germain} & \# \text{ Twin} & c=1.32032 \newline \hline 22& 421 & 374 & 470 & 522 & 494 \newline \hline 23& 346 & 343 & 494 & 456 & 452\newline \hline 24& 342 & 315 & 428 & 419 & 426\newline \hline 25& 259 & 291 & 407 & 451 & 384 \end{array}$$
