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 A006450 Prime-indexed primes: primes with prime subscripts. (Formerly M2477) 236
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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] 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. 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 EXAMPLE a(5) = 31 because a(5) = p(p(5)) = p(11) = 31. 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} ] PROG (MAGMA) [ NthPrime(NthPrime(n)): n in [1..51] ]; // Jason Kimberley, Apr 02 2010 (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 (Haskell) a006450 = a000040 . a000040 a006450_list = map a000040 a000040_list -- Reinhard Zumkeller, Jan 12 2013 (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 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. Sequence in context: A309438 A309439 A108542 * A085918 A267094 A336372 Adjacent sequences:  A006447 A006448 A006449 * A006451 A006452 A006453 KEYWORD easy,nice,nonn AUTHOR Jeffrey Shallit, Nov 25 1975 STATUS approved

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Last modified May 23 23:00 EDT 2022. Contains 353993 sequences. (Running on oeis4.)