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A006450
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Prime-indexed primes: primes with prime subscripts.
(Formerly M2477)
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252
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Eric Weisstein's World of Mathematics, Prime formulas, see Cipolla formula.
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FORMULA
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a(n) > n*(log(n))^2, as prime(n) > n*log(n) by Rosser's theorem. - Jonathan Sondow, Jul 11 2012
Sum_{n>=1} 1/a(n) is in the interval (1.04299, 1.04365) (Bayless et al., 2013). - Amiram Eldar, Oct 15 2020
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EXAMPLE
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a(5) = 31 because a(5) = p(p(5)) = p(11) = 31.
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MAPLE
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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
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MATHEMATICA
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Table[ Prime[ Prime[ n ] ], {n, 100} ]
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PROG
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(Magma) [ NthPrime(NthPrime(n)): n in [1..51] ]; // Jason Kimberley, Apr 02 2010
(Haskell)
a006450 = a000040 . a000040
a006450_list = map a000040 a000040_list
(Python)
from sympy import prime
def a(n): return prime(prime(n))
(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
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CROSSREFS
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Cf. A000040, A007821, A038580, A049090, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.
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.
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KEYWORD
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easy,nice,nonn
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AUTHOR
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STATUS
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approved
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