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 A007097 Primeth recurrence: a(n+1) = a(n)-th prime. (Formerly M0734) 251
 1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey, Jun 26 2005 Lubomir Alexandrov informs me that he studied this sequence in his 1965 notebook. - N. J. A. Sloane, May 23 2008 a(n) is the Matula-Goebel number of the rooted path tree on n+1 vertices. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012 Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - Thomas Ordowski, Mar 26 2015 This conjecture was disproved, see the MathOverflow link. - Ahmad J. Masad, Apr 17 2020 REFERENCES Lubomir Alexandrov, unpublished notes, circa 1960. L. Longeri, Towards understanding nature and the aesthetics of prime numbers, https://www.longeri.org/prime/nature.html [Broken link, but leave the URL here for historical reasons] N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math.NT/9811096, 1998. Lubomir Alexandrov, "The Eratosthenes Progression p(k+1)=π^{-1}(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19. Available at arXiv:math/0105154 [math.NT], 2001. Lubomir Alexandrov, Prime Number Sequences And Matrices Generated By Counting Arithmetic Functions, Communications of the Joint Institute of Nuclear Research, E5-2002-55, Dubna, 2002. J. Awbrey, Riffs and Rotes 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. Peter R. Cappello, A Note on a Bijection between Natural Numbers and Rooted Trees, 4th SIAM Conference on Discrete Mathematics, June 1988.  See section 3 set S codes of paths (codes are per Matula-Goebel). M. Deléglise, Computation of large values of pi(x) N. Fernandez, An order of primeness, F(p) N. Fernandez, An order of primeness [cached copy, included with permission of the author] MathOverflow, Thomas Ordowski conjecture at OEIS sequence A007097, 2020. N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98). R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992 FORMULA A049084(a(n+1)) = a(n). - Reinhard Zumkeller, Jul 14 2013 a(n)/a(n-1) ~ log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 26 2015 a(n) = prime^{[n]}(1), with the prime function prime(k) = A000040(k), with a(0) = 1. See the name and the programs. - Wolfdieter Lang, Apr 03 2018 MAPLE seq((ithprime@@n)(1), n=0..10); # Peter Luschny, Oct 16 2012 MATHEMATICA NestList[Prime@# &, 1, 16] (* Robert G. Wilson v, May 30 2006 *) PROG (PARI) print1(p=1); until(, print1(", "p=prime(p)))  \\ M. F. Hasler, Oct 09 2011 (Haskell) a007097 n = a007097_list !! n a007097_list = iterate a000040 1  -- Reinhard Zumkeller, Jul 14 2013 (GAP) P:=Filtered([1..60000], IsPrime);; a:=[1];; for n in [2..10] do a[n]:=P[a[n-1]]; od; a; # Muniru A Asiru, Dec 22 2018 CROSSREFS Row 1 of array A114537. Left edge of tree A227413, right edge of A246378. Cf. A000040, A000720, A049076-A049081, A049084, A057450, A109301, A131842, A006450. Cf. A078442, A109082 (left inverses). Subsequence of A245823. Sequence in context: A090709 A112279 A130166 * A173422 A132745 A124538 Adjacent sequences:  A007094 A007095 A007096 * A007098 A007099 A007100 KEYWORD nonn,hard,nice,changed AUTHOR EXTENSIONS a(15) corrected and a(16)-a(17) added by Paul Zimmermann a(18)-a(19) found by David Baugh using a program by Xavier Gourdon and Andrey V. Kulsha, Oct 25 2007 a(20)-a(21) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 02 2011 a(22) from Henri Lifchitz, Oct 14 2014 a(23) from David Baugh using Kim Walisch's primecount, May 16 2016 STATUS approved

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Last modified September 18 23:30 EDT 2020. Contains 337175 sequences. (Running on oeis4.)