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Higher-order prime numbers
Higher-order prime numbers, also called superprime numbers (super-prime numbers, super-primes or superprimes), a subsequence of the prime numbers, are the primes that occupy prime-numbered positions within the sequence of all prime numbers. They are also called prime-indexed primes.
a (n) = p pn |
- {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, ...}
pi |
i |
p pi |
Contents
- 1 Asymptotic behavior of the superprimes
- 2 Superprime gaps
- 3 Harmonic series of the superprimes
- 4 Higher-order superprimes
- 5 Sequences
- 6 See also
- 7 Notes
- 8 References
- 9 External links
Asymptotic behavior of the superprimes
Broughan and Barnett[1] show that there are
-
+ Ox (log x) 2 x log log x (log x) 3
x |
k |
pk ∼ k log k |
k = n log n |
p pn |
-
p pn ∼ (n log n) log (n log n) ∼ (n log n) (log n + log log n) ∼ n (log n) 2,
n |
p pn |
-
∼n p pn
∼n n (log n) 2
,1 (log n) 2
in agreement with Broughan and Barnett.
Superprime gaps
A073131 Superprime gaps:a (n) = p pn + 1 − p pn |
pi |
i |
- {2, 6, 6, 14, 10, 18, 8, 16, 26, 18, 30, 22, 12, 20, 30, 36, 6, 48, 22, 14, 34, 30, 30, 48, 38, 16, 24, 12, 18, 92, 30, 34, 24, 62, 18, 42, 48, 24, 40, 32, 24, 66, 18, 30, ...}
Harmonic series of the superprimes
The harmonic series of the superprimes (series of the reciprocals of the superprimes) converges to
-
S2 = ∞∑ i = 1
=1 p pi
+1 p2
+1 p3
+1 p5
+1 p7
+1 p11
+1 p13
+1 p17
+ ⋯ =1 p19
+1 3
+1 5
+1 11
+1 17
+1 31
+1 41
+1 59
+ ⋯ = ?1 67
Higher-order superprimes
One can also define “higher-order” primeness much the same way, and obtain analogous sequences of primes. (Fernandez 1999)
Order of primeness
LetS ( p) = S ( pk ) = k |
p |
a (n) |
pn = 1 + m |
m |
S (S ( ⋯ S ( pn ) ⋯ )) |
m |
S |
F (n) = 0 |
n |
-
:F ( p) = 1
is prime but not primeth prime (p
not prime);S ( p) -
:F ( p) = 2
is primeth prime but not primeth primeth prime (p
not prime);S (S ( p)) -
:F ( p) = 3
is primeth primeth prime but not primeth primeth primeth prime (p
not prime);S (S (S ( p))) - ...
n |
- {1, 2, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,, ...}
“Order of primeness” − “order of compositeness”
Since the nonprimes have order of primeness 0 and the noncomposites have order of compositeness 0, if we subtract the order of compositeness from the order of primeness we get a “negative order of primeness” (negation of order of compositeness) for the composite numbers, and
- primes give a positive value (order of primeness),
- 1 gives zero (both order of primeness and order of compositiveness being zero),
- composites give a negative value (negation of order of compositiveness).
n |
a ( p (n)) = a (n) + 1 |
n |
p (n) |
a (n) = 0 |
n |
- {0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, ...}
n |
a (c (n)) = a (n) + 1 |
n |
c (n) |
a (n) = 0 |
n |
n |
- {0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 1, 3, 1, 0, 2, 3, 4, 1, 3, 0, 1, 0, 2, ...}
A?????? “Order of primeness” − “order of compositeness.”
- {0, 1, 2, −1, 3, −1, 1, −1, −2, −1, 4, −2, 1, −1, −2, −3, 2, −2, 1, −1, −3, −1, 1, −2, −3, − 4, −1, −3, 1, −1, 5, −2, ...}
Superprimes with order of primeness at least k
p |
k |
Asymptotic behavior of the superprimes with order of primeness at least k
Using the asymptotic behavior of thek |
p pk ∼ k (log k) 2 |
k = n log n |
p ppn |
-
p ppn ∼ (n log n) (log (n log n)) 2 ∼ (n log n) (log n + log log n) 2 ∼ n (log n) 3,
n |
p ppn |
-
∼n p ppn
∼n n (log n) 3
.1 (log n) 3
By induction, we obtain
thus giving the asymptotic density
Harmonic series of the superprimes with order of primeness at least k
The harmonic series of the superprimes (series of the reciprocals of the superprimes) with order of primeness at leastk |
Superprimes with order of primeness equal to k
p |
k |
Harmonic series of the superprimes with order of primeness equal to k
The harmonic series of the superprimes (series of the reciprocals of the superprimes) with order of primeness equal tok |
sk = Sk − Sk + 1 |
k ≥ 2 |
S1 |
n log n |
S2 |
s1 = S1 − S2 |
n log n |
Sequences
A175247 Primes (A000040) with noncomposite (A008578) subscripts. (Same as A006450, prepended withp (1) = 2 |
- {2, 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, ...}
A018252 The nonprime numbers (unit 1 together with the composite numbers, A002808). (Order of primeness is 0.)
See also
Notes
- ↑ 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.
- ↑ In the OEIS, all the
’s are incremented by 1, so that “ ≥ ” are replaced by “ > ”. (Why?)k - ↑ In the OEIS, all the
’s are incremented by 1. (Why?)k
References
- 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).
External links
- Michael R. Mirzayanov, A Russian programming contest problem related to the work of Dressler and Parker, 2001.