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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.
A006450 Primes with prime subscripts: 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, ...}
That is, if pi i p
pi
Contents
- 1 Asymptotic behavior of the superprimes
- 2 Superprime gaps
- 3 Harmonic series of the superprimes
- 4 Higher-order superprimes
- 4.1 Order of primeness
- 4.1.1 “Order of primeness” − “order of compositeness”
- 4.2 Superprimes with order of primeness at least k
- 4.2.1 Asymptotic behavior of the superprimes with order of primeness at least k
- 4.2.2 Harmonic series of the superprimes with order of primeness at least k
- 4.3 Superprimes with order of primeness equal to k
- 4.3.1 Harmonic series of the superprimes with order of primeness equal to k
- 4.1 Order of primeness
- 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
superprimes up to 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,
thus giving the asymptotic density (there being 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
Let S ( p) = S ( pk ) = k p a (n) pn = 1 + m m S (S ( ⋯ S ( pn ) ⋯ )) m S F (n) = 0 n
-
F ( p) = 1
:p
is prime but not primeth prime (S ( p)
not prime); -
F ( p) = 2
:p
is primeth prime but not primeth primeth prime (S (S ( p))
not prime); -
F ( p) = 3
:p
is primeth primeth prime but not primeth primeth primeth prime (S (S (S ( p)))
not prime); - ...
A049076 Order of primeness: number of steps in the prime index chain for the 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).
A078442 “Order of primeness” of 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, ...}
A?????? “Order of compositeness” of 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
k | Sequence | A-number |
---|---|---|
1 | {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, ...} | A000040 |
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, ...} | A006450 |
3 | {5, 11, 31, 59, 127, 179, 277, 331, 431, 599, 709, 919, 1063, 1153, 1297, 1523, 1787, 1847, 2221, 2381, 2477, 2749, 3001, 3259, 3637, 3943, 4091, 4273, 4397, ...} | A038580 |
4 | {11, 31, 127, 277, 709, 1063, 1787, 2221, 3001, 4397, 5381, 7193, 8527, 9319, 10631, 12763, 15299, 15823, 19577, 21179, 22093, 24859, 27457, 30133, 33967, ...} | A049090 |
5 | {31, 127, 709, 1787, 5381, 8527, 15299, 19577, 27457, 42043, 52711, 72727, 87803, 96797, 112129, 137077, 167449, 173867, 219613, 239489, 250751, 285191, ...} | A049203 |
6 | {127, 709, 5381, 15299, 52711, 87803, 167449, 219613, 318211, 506683, 648391, 919913, 1128889, 1254739, 1471343, 1828669, 2269733, 2364361, 3042161, ...} | A049202 |
7 | {709, 5381, 52711, 167449, 648391, 1128889, 2269733, 3042161, 4535189, 7474967, 9737333, 14161729, 17624813, 19734581, 23391799, 29499439, 37139213, ...} | A057849 |
8 | {5381, 52711, 648391, 2269733, 9737333, 17624813, 37139213, 50728129, 77557187, 131807699, 174440041, 259336153, 326851121, 368345293, 440817757, ...} | A057850 |
9 | {52711, 648391, 9737333, 37139213, 174440041, 326851121, 718064159, 997525853, 1559861749, 2724711961, 3657500101, 5545806481, 7069067389, ...} | A057851 |
10 | {648391, 9737333, 174440041, 718064159, 3657500101, 7069067389, 16123689073, 22742734291, 36294260117, 64988430769, 88362852307, 136395369829, ...} | A057847 |
11 | {9737333, 174440041, 3657500101, 16123689073, 88362852307, 175650481151, 414507281407, 592821132889, 963726515729, 1765037224331, 2428095424619, ...} | A058332 |
12 | {174440041, 3657500101, 88362852307, 414507281407, 2428095424619, 4952019383323, 12055296811267, 17461204521323, 28871271685163, 53982894593057, ...} | A093047 |
Asymptotic behavior of the superprimes with order of primeness at least k
Using the asymptotic behavior of the k 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,
thus giving the asymptotic density (there being 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 least k
Superprimes with order of primeness equal to k
k | Sequence | A-number |
---|---|---|
1 | {2, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 181, 193, 197, 199, 223, 227, 229, 233, 239, ...} | A007821 |
2 | {3, 17, 41, 67, 83, 109, 157, 191, 211, 241, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1087, 1171, 1201, ...} | A049078 |
3 | {5, 59, 179, 331, 431, 599, 919, 1153, 1297, 1523, 1847, 2381, 2477, 2749, 3259, 3637, 3943, 4091, 4273, 4549, 5623, 5869, 6113, 6661, 6823, 7607, 7841, ...} | A049079 |
4 | {11, 277, 1063, 2221, 3001, 4397, 7193, 9319, 10631, 12763, 15823, 21179, 22093, 24859, 30133, 33967, 37217, 38833, 40819, 43651, 55351, 57943, 60647, ...} | A049080 |
5 | {31, 1787, 8527, 19577, 27457, 42043, 72727, 96797, 112129, 137077, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, ...} | A049081 |
6 | {127, 15299, 87803, 219613, 318211, 506683, 919913, 1254739, 1471343, 1828669, 2364361, 3338989, 3509299, 4030889, 5054303, 5823667, 6478961, 6816631, ...} | A058322 |
7 | {709, 167449, 1128889, 3042161, 4535189, 7474967, 14161729, 19734581, 23391799, 29499439, 38790341, 56011909, 59053067, 68425619, 87019979, 101146501, ...} | A058324 |
8 | {5381, 2269733, 17624813, 50728129, 77557187, 131807699, 259336153, 368345293, 440817757, 563167303, 751783477, 1107276647, 1170710369, 1367161723, ...} | A058325 |
9 | {52711, 37139213, 326851121, 997525853, 1559861749, 2724711961, 5545806481, 8012791231, 9672485827, 12501968177, 16917026909, 25366202179, ...} | A058326 |
10 | {648391, 718064159, 7069067389, 22742734291, 36294260117, 64988430769, 136395369829, 200147986693, 243504973489, 318083817907, 435748987787, ...} | A058327 |
11 | {9737333, 16123689073, 175650481151, 592821132889, 963726515729, 1765037224331, 3809491708961, 5669795882633, 6947574946087, 9163611272327, ...} | A058328 |
12 | {174440041, 414507281407, 4952019383323, 17461204521323, 28871271685163, 53982894593057, 119543903707171, 180252380737439, 222334565193649, ...} | A093046 |
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 to k 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 with p (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
- Higher-order composite numbers
- Order of compositeness
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
k
’s are incremented by 1, so that “ ≥ ” are replaced by “ > ”. (Why?) - ↑ In the OEIS, all the
k
’s are incremented by 1. (Why?)
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.