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$127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime number and and so its rank ($3$) is also a prime. And finally $3$ is the $2$nd prime so its rank is also a prime... Is there a name for primes whose rank (index in the prime series) is also a prime? That is:

If $a_i$ is the $i$th prime number, then $i$ is a also a prime.

How about numbers like $127$ where going down ranks of ranks always produce prime numbers (down to rank $2$ of course)? The first $11$ :-) primes in this (infinite) series would be:

$3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, \dotsc$

Many thanks!

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2 Answers 2

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This sequence is well-know at OEIS, namely sequence A007097, where one can find a lot of information and references: $$ 1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041$$ The name is "Primeth recurrence": $a(n+1) = a(n)$-th prime.

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  • $\begingroup$ Thanks! That is exactly the info I was looking for... $\endgroup$
    – Pan Pap
    Aug 24, 2017 at 13:41
  • $\begingroup$ Done! Thanks once more... $\endgroup$
    – Pan Pap
    Aug 24, 2017 at 16:07
  • $\begingroup$ Does anyone here know if it's a finite sequence or if those are just the last ones someone bothered to calculate ? $\endgroup$ Aug 24, 2017 at 16:34
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    $\begingroup$ It's clearly infinite - given a(n) = p you can always calculate a(n+1) as being the p-th prime. $\endgroup$ Aug 24, 2017 at 16:50
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    $\begingroup$ Another relevant OEIS sequence: oeis.org/A288469 $\endgroup$ Aug 24, 2017 at 20:11
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Sloane's OEIS simply calls them "prime-indexed primes."

Some more common notation would be $p_i$ for the $i$th prime. Then we have the prime counting function $\pi(p_i) = i$ (this function is defined for all positive numbers).

Using these notations, we can write the sequence $$1, p_1, p_{p_1}, p_{p_{p_1}}, p_{p_{p_{p_1}}}, \ldots$$

This is Wilson's primeth recurrence.

Although both the sequence of prime-indexed primes and Wilson's primeth recurrence are infinite sequences, the former can be said to be less "exclusive" than the latter. If we iterate the function $f(n) = p_{\pi(n)}$, we'll find that all prime-indexed primes eventually reach a nonprime, but only those in Wilson's primeth recurrence reach $1$ without any composite numbers along the way.

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