Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers.

The subsequence begins

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... (sequence A006450 in the OEIS).

That is, if p(i) denotes the ith prime number, the numbers in this sequence are those of the form p(p(i)).

Dressler & Parker (1975) used a computer-aided proof (based on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof relies on a result resembling Bertrand's postulate, stating that (after the larger gap between super-primes 5 and 11) each super-prime number is less than twice its predecessor in the sequence.

Broughan & Barnett (2009) show that there are

${\displaystyle {\frac {x}{(\log x)^{2))}+O\left({\frac {x\log \log x}{(\log x)^{3))}\right)}$

super-primes up to x. This can be used to show that the set of all super-primes is small.

One can also define "higher-order" primeness much the same way and obtain analogous sequences of primes (Fernandez 1999).

A variation on this theme is the sequence of prime numbers with palindromic prime indices, beginning with

3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, ... (sequence A124173 in the OEIS).

## References

• Bayless, Jonathan; Klyve, Dominic; Oliveira e Silva, Tomás (2013), "New bounds and computations on prime-indexed primes", Integers, 13: A43:1–A43:21, MR 3097157
• Broughan, Kevin A.; Barnett, A. Ross (2009), "On the subsequence of primes having prime subscripts", Journal of Integer Sequences, 12, article 09.2.3.
• Dressler, Robert E.; Parker, S. Thomas (1975), "Primes with a prime subscript", Journal of the ACM, 22 (3): 380–381, doi:10.1145/321892.321900, MR 0376599.
• Fernandez, Neil (1999), An order of primeness, F(p).