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$2$ is the $1$st prime.

$3$ is the $2$nd prime.

$5$ is the $3$rd prime.

$11$ is the $5$th prime.

$31$ is the $11$th prime.

...

I'm just wondering if there is a name for this integer sequence, is it on OEIS?. I find it hard to find a suitable search term for it.

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    $\begingroup$ oeis.org/… $\endgroup$ Commented Nov 18, 2017 at 19:20
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    $\begingroup$ I'd imagine there is no name nor search term for it because it's absolutely meaningless, both theoretically and practically. $\endgroup$
    – user436658
    Commented Nov 18, 2017 at 19:20
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    $\begingroup$ The formula is $\large p_{p_n}$, but I never heard a name for it. And why is this theoretically meaningless ? I once asked whether the sum of the reciprocals of those numbers converges. It is a quite natural next step. We have the primes, and then, we only consider primes $p_n$ , for which the index $n$ is prime. But how should we call those numbers ? Does anyone have a good idea ? $\endgroup$
    – Peter
    Commented Nov 18, 2017 at 19:44
  • $\begingroup$ @Peter. By the prime number theorem $p_n \sim n \log n$ thus $$\sum_n \frac{1}{p_{p_n}} \le C\sum_n \frac{1}{p_n\log p_n} \le C_2 \sum_n \frac{1}{(n\log n) \log( n\log n)} \le C_3 \sum_n \frac{1}{n \log^2 n} < \infty$$ You can create thousands of such meaningless and useless functions/sequences in number theory. In the same way there is no name for the antiderivative of $\frac{\exp(\sin(x^{1/2}))}{\tan x}$. $\endgroup$
    – reuns
    Commented Nov 18, 2017 at 19:48
  • $\begingroup$ oeis.org/… wolframalpha.com/input/… $\endgroup$
    – reuns
    Commented Nov 18, 2017 at 19:54

1 Answer 1

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The numbers in your sequence form OEIS A007097, i.e., the recurrence denoted by $a(n)$ such that:

$$ a(n+1) = a(n)^{\rm th} \rm \ \ prime. $$

This is called the primeth recurrence.

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