Super-prime
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. In other words, if you matched prime numbers with ordinal numbers, starting with prime number 2 matched with ordinal number 1, the primes matched with prime ordinal numbers are the super primes.
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(n) denotes the nth prime number, the numbers in this sequence are those of the form p(p(n)).
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| p(n) | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 |
| p(p(n)) | 3 | 5 | 11 | 17 | 31 | 41 | 59 | 67 | 83 | 109 | 127 | 157 | 179 | 191 | 211 | 241 | 277 | 283 | 331 | 353 |
(Dressler Parker) 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) show that there are
- [math]\displaystyle{ \frac{x}{(\log x)^2} + O\left(\frac{x\log\log x}{(\log x)^3}\right) }[/math]
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
References
- Bayless, Jonathan; Klyve, Dominic; Oliveira e Silva, Tomás (2013), "New bounds and computations on prime-indexed primes", Integers 13: A43:1–A43:21, http://digitalcommons.cwu.edu/cgi/viewcontent.cgi?article=1004&context=math
- Broughan, Kevin A.; Barnett, A. Ross (2009), "On the subsequence of primes having prime subscripts", Journal of Integer Sequences 12: article 09.2.3, http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html.
- 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.
- Fernandez, Neil (1999), An order of primeness, F(p), http://borve.org/primeness/FOP.html.
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