What is the natural density of hyper prime numbers?

Question

What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et cetera.

The common soul of these generalized prime numbers is their irreducibility to the smaller numbers with respect to the given hyperoperator; the same property that distinguishes usual primes from other natural numbers. In fact, such numbers don't arise as a non-trivial combination of smaller building blocks. The underlying combination method could be multiplication, exponentiation or any other hyperoperation.

Definition 1. We call addition, multiplication, exponentiation, tetration,... , the $1$-hyperoperator, $2$-hyperoperator, $3$-hyperoperator, $4$-hyperoperator, ... respectively and use the $*_n$ notation to denote the $n$-hyperoperator for $n\geq 1$.

Definition 2. For any $n\geq 1$, the natural number $p>1$ is called a $n$-hyperprime number if there are no $i,j<p$ such that $p=i*_{n}j$.

Remark 1. Note that according to the definition, there are no $1$-hyperprime numbers, simply because every natural number greater than $1$ arises as the addition of two smaller natural numbers. $2$-hyperprime numbers are exactly the usual prime numbers and the set of $3$-hyperprime numbers consists of usual primes and numbers like $12$ which don't arise as the exponentiation of two smaller numbers. In general, if $p>1$ is a $m$-hyperprime number for some $m\geq 1$ then it is $n$-hyperprime for any $n>m$.

Remark 2. It is intuitively clear that almost all natural numbers are hyperprime in some sense, simply because the hyperoperator $*_n$ gets violently powerful when $n$ grows. Thus for almost any natural number $p>1$ one may find sufficiently large $n$ such that even the smallest combinations of natural numbers like $2*_{n}3$ and $3*_{n}2$ exceed $p$. So based on the increasing nature of hyperoperators, $p$ could not arise as the $*_n$ combination of any smaller numbers and is a $n$-hyperprime number. The only (magical?) exception is the eternally composite number $4$ which is NOT $n$-hyperprime for any $n\geq 1$ because $\forall n\geq 1~~~4=2*_{n}2$.

Now the first question is: "How many hyperprimes of any type are there?" We know that there are no $1$-hyperprimes. Also, there are very few $2$-hyperprime numbers in the sense that the natural density of the set of prime numbers in $\mathbb{N}$ is $0$. Roughly speaking, the usual prime numbers are so rare among natural numbers that one may assume that there is almost no such number!

But as stated, there is a promising increasing trend among hyperprime numbers of higher order. When $n$ gets larger and larger, the $n$-hyperoperator skips more and more numbers in its range (while receiving non-trivial input) leaving them as $n$-hyperprimes. Consequently, the corresponding set of $n$-hyperprime numbers gets rapidly larger by varying $n$. So one may cover almost the entire set of natural numbers this way eventually. Thus it is natural to ask:

Question 1. Where is the first place that almost all natural numbers are prime with respect to a hyperoperator?

Precisely, what is the least natural number $n$ such that the set of all $n$-hyperprime numbers is of natural density $1$ in $\mathbb{N}$?

What about the other values in between? Is there a natural number $n$ such that the natural density of the set of all $n$-hyperprime numbers lies in the interval $(0,1)$? (And so we have a balanced number of prime numbers, neither too few nor too many!)

In order to answer the above question one may need to figure out the distribution of hyperprime numbers in $\mathbb{N}$ which in the case of usual prime numbers is given by Prime Number Theorem that provides the $\frac{k}{\ln(k)}$ estimation for the number of $2$-hyperprime numbers in the interval $[1, k]$.

Question 2. What is the analogy of Prime Number Theorem for $n$-hyperprimes in general? In other words, what is the growth rate of the $n$-hyperprime counting function $\pi_{n}(x)$ which assigns the number of $n$-hyperprime numbers $\leq x$ to the real number $x$? For $n=2$ it is known to be $\frac{x}{\ln (x)}$.

Finally, one may ask about the analogy of Fundamental Theorem of Arithmetic in the case of hyperprimes. Although, we are not dealing with a commutative hyperoperator for $n\geq 3$ but it is still meaningful and interesting to wonder whether every natural number has a unique representation in terms of $n$-hyper combination of $n$-hyperprime numbers or not? For instance, $8^{9^{10}}$ is a natural number. While $8=2^3$ and $9=3^2$ aren't $3$-hyperprime but this number has a representation $2^{3^{21}}$ via $3$-hyperprime numbers $2, 3, 21$. Does such a representation exist for every other number? Is it unique?

Question 3. Does a variant of Fundamental Theorem of Arithmetic hold for any $n$-hyperoperator using the mentioned notion of $n$-hyperprimeness?

Precisely, is it true that for every natural numbers $n>1$ and $k>1$, there is $s\geq 1$ and a unique sequence of $n$-hyperprime numbers $p_1, \cdots, p_{s}$ such that $k=p_s *_{n}(\cdots (p_3 *_{n}(p_2 *_{n} p_1)))$?

Answer 1

The answer to Question 1 is that the $3$-hyperprimes have density $1$.

For any $k\ge 2$ there are only $\lfloor n^{1/k}\rfloor$ numbers are of the form $a^k$ between $1$ and $n$. When $k>\log_2(n)$ (i.e. $2^k>n$) only $1$ is of that form. Hence there are at most $n^{1/2}+\cdots+n^{1/\log_2(n)}=O(n^{1/2}\log(n))$ $3$-hypercomposites from $1$ to $n$.

Answer 2

I can answer question 3 - I've written a paper that covers it ( https://www.preprints.org/manuscript/201902.0176/v1 ).

It turns out that for the 3-operation, then yes, every number has a unique representation as a sequence of 3-operations, where every number in that sequence is 3-prime.

However, for the 4-operation and higher, the unique representation only guarantees that the last number in the sequence of operations is n-prime ($p_1$ in your notation).