# Super-prime

Super-prime numbers (also known as "higher order primes") 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, … (sequence A006450 in 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 and Barnett[1] show that there are

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

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 indices, beginning with

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

## References

1. ^ Kevin A. Broughan and A. Ross Barnett, On the Subsequence of Primes Having Prime Subscripts, Journal of Integer Sequences 12 (2009), article 09.2.3.