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A sequence of objects in a set $S$ is, by definition, a function $f$ from the set of natural numbers $\mathbb{N}=\{1,2,\ldots\}$ to $S$.

I wanted to see whether the following function makes sense to say that we have defined a sequence of prime numbers.

Let $f:\mathbb{N}\rightarrow \mathbb{N}$, $f(n)$ is $n$-th prime number. Here prime numbers are taken from $2$, and written in increasing order, and then enumerated as first, second, etc.

Does it make sense to say that this is a sequence of prime numbers? If yes, can we assert that $f(n)$ can not be expressed by any formula in $n$? (As we know there are infinitely many primes, but we do not an algebraic expression in $n$ which expresses the $n$-th prime number, am I right?)

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    $\begingroup$ You are right, this is a well-defined sequence, but indeed it has no closed form. $\endgroup$ Commented Sep 2, 2020 at 10:20
  • $\begingroup$ May I get to know what it closed form? $\endgroup$ Commented Sep 2, 2020 at 10:22
  • $\begingroup$ I mean that, as you said, you cannot express the $n-$th term in terms of $n$. $\endgroup$ Commented Sep 2, 2020 at 10:23
  • $\begingroup$ Can you precise what a formula in $n$ is? Enumerating the primes is primitive recursive. $\endgroup$ Commented Sep 2, 2020 at 10:27
  • $\begingroup$ I mean, since currently exact values of primes are known up to a certain limit, but beyond it, we do not know the prime number; we know only that it exists. This is because, there is no expression in terms of $n$ and other known integers, real numbers, which give $n$-th prime number. $\endgroup$ Commented Sep 2, 2020 at 10:31

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Does it make sense to say that this is a sequence of prime numbers?

Yes, such a sequence makes perfect sense.

If yes, can we assert that $f(n)$ can not be expressed by any formula in $n$?

This all depends on what you call a formula. For example you can find a primitive recursive function that enumerates the primes. See this article for details.

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