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 show that there are
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
- 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.
- 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).
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