Super-prime

For the computer program, see SuperPrime.

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, 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 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 prime indices, beginning with

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

References

↑ 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) .

External links

A Russian programming contest problem related to the work of Dressler and Parker

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

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