A006450 Prime-indexed primes: primes with prime subscripts.

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, 1031, 1063, 1087, 1153, 1171, 1201, 1217, 1297, 1409, 1433, 1447, 1471

Offset: 1

Author

Jeffrey Shallit, Nov 25 1975

Keywords

• easy
• nice
• nonn

Comments

A000040 = A006450 U A007821. - Juri-Stepan Gerasimov, Sep 24 2009

Subsequence of A175247 (primes (A000040) with noncomposite (A008578) subscripts), a(n) = A175247(n+1). - Jaroslav Krizek, Mar 13 2010

Primes p such that p and pi(p) are both primes. - Juri-Stepan Gerasimov, Jul 14 2011

Sum_{n>=1} 1/a(n) converges. In fact, Sum_{n>N} 1/a(n) < 1/log(N), by the integral test. - Jonathan Sondow, Jul 11 2012

The number of such primes not exceeding x > 0 is pi(pi(x)). I conjecture that the sequence a(n)^(1/n) (n = 1,2,3,...) is strictly decreasing. This is an analog of the Firoozbakht conjecture on primes. - Zhi-Wei Sun, Aug 17 2015

Limit_{n->infinity} a(n)/(n*(log(n))^2) = 1. Proof: By Cipolla's asymptotic formula, prime(n) ~ L(n) + R(n), where L(n)/n = log(n) + log(log(n)) - 1 and R(n)/n decreases logarithmically to 0. Hence, for large n, a(n) = prime(prime(n)) ~ L(L(n)+R(n)) + R(L(n)+R(n)) = n*(log(n))^2 + r(n), where r(n) grows as O(n*log(n)*log(log(n))). The rest of the proof is trivial. The convergence is very slow: for k = 1,2,3,4,5,6, sqrt(a(10^k)/10^k)/log(10^k) evaluates to 2.055, 1.844, 1.695, 1.611, 1.545, and 1.493, respectively. - Stanislav Sykora, Dec 09 2015

Examples

			a(5) = 31 because a(5) = p(p(5)) = p(11) = 31.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• J. S. Kimberley, Table of n, a(n) for n = 1..100000
• R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
• Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.
• K. A. Broughan and A. R. Barnett, On the subsequence of primes having prime subscripts, JIS 12 (2009) 09.2.3.
• Paul Cooijmans, Numbers.
• Paul Cooijmans, Short Test For Genius.
• R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM 22 (1975) 380-381.
• N. Fernandez, An order of primeness, F(p)
• N. Fernandez, An order of primeness [cached copy, included with permission of the author]
• N. Fernandez, More terms of this and other sequences related to A049076.
• A. B. Frizell, The permutations of the natural numbers can not be well ordered, Bull. Amer. Math. Soc. 22 (1915), no. 2, 71-73.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
• Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
• J. Shallit, Letter to N. J. A. Sloane, Oct. 1975
• Eric Weisstein's World of Mathematics, Prime formulas, see Cipolla formula.

Crossrefs

Primes for which A049076 > 1.

Cf. A000040, A007821, A038580, A049090, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.

Cf. A185723 and A214296 for numbers and primes that are sums of distinct a(n); cf. A213356 and A185724 for those that are not.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270795, A270796, A102616.

Programs

Haskell

a006450 = a000040 . a000040
a006450_list = map a000040 a000040_list
-- Reinhard Zumkeller, Jan 12 2013

Magma

[ NthPrime(NthPrime(n)): n in [1..51] ]; // Jason Kimberley, Apr 02 2010

Maple

seq(ithprime(ithprime(i)),i=1..50); # Uli Baum (Uli_Baum(AT)gmx.de), Sep 05 2007
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

Mathematica

Table[ Prime[ Prime[ n ] ], {n, 100} ]

PARI

i=0;forprime(p=2,1e4,if(isprime(i++),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

PARI

a=vector(10^3,n,prime(prime(n))) \\ Stanislav Sykora, Dec 09 2015

Python

from sympy import prime
def a(n): return prime(prime(n))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Aug 11 2021

Python

# much faster version for initial segment of sequence
from sympy import nextprime, isprime
def aupton(terms):
    alst, p, pi = [], 2, 1
    while len(alst) < terms:
        if isprime(pi): alst.append(p)
        p, pi = nextprime(p), pi+1
    return alst
print(aupton(10000)) # Michael S. Branicky, Aug 11 2021

Formula

a(n) = prime(prime(n)) = A000040(A000040(n)). - Juri-Stepan Gerasimov, Sep 24 2009

a(n) > n*(log(n))^2, as prime(n) > n*log(n) by Rosser's theorem. - Jonathan Sondow, Jul 11 2012

a(n)/log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 30 2015

Sum_{n>=1} 1/a(n) is in the interval (1.04299, 1.04365) (Bayless et al., 2013). - Amiram Eldar, Oct 15 2020