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%I M0734 #208 Jan 31 2023 09:34:20
%S 1,2,3,5,11,31,127,709,5381,52711,648391,9737333,174440041,3657500101,
%T 88362852307,2428095424619,75063692618249,2586559730396077,
%U 98552043847093519,4123221751654370051,188272405179937051081,9332039515881088707361,499720579610303128776791,28785866289100396890228041
%N Primeth recurrence: a(n+1) = a(n)-th prime.
%C 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
%C _Lubomir Alexandrov_ informs me that he studied this sequence in his 1965 notebook. - _N. J. A. Sloane_, May 23 2008
%C 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
%C Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - _Thomas Ordowski_, Mar 26 2015
%D Lubomir Alexandrov, unpublished notes, circa 1960.
%D 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]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Lubomir Alexandrov, <a href="https://arxiv.org/abs/math/9811096">On the nonasymptotic prime number distribution</a>, arXiv:math.NT/9811096, 1998.
%H 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 <a href="https://arxiv.org/abs/math/0105154">arXiv:math/0105154 [math.NT]</a>, 2001.
%H Lubomir Alexandrov, <a href="http://www1.jinr.ru/Preprints/2002/055(E5-2002-55).pdf">Prime Number Sequences And Matrices Generated By Counting Arithmetic Functions</a>, Communications of the Joint Institute of Nuclear Research, E5-2002-55, Dubna, 2002.
%H J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>
%H R. G. Batchko, <a href="https://arxiv.org/abs/1405.2900">A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes</a>, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
%H Peter R. Cappello, <a href="http://sites.cs.ucsb.edu/~cappello/papers/1988SiamDM.html">A Note on a Bijection between Natural Numbers and Rooted Trees</a>, 4th SIAM Conference on Discrete Mathematics, June 1988. See section 3 set S codes of paths (codes are per Matula-Goebel).
%H M. Deléglise, <a href="http://algo.inria.fr/seminars/sem95-96/deleglise.pdf">Computation of large values of pi(x)</a>
%H N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H Robert G. Wilson v, <a href="/A007097/a007097.pdf">Letter to N. J. A. Sloane, Sep. 1992</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A049084(a(n+1)) = a(n). - _Reinhard Zumkeller_, Jul 14 2013
%F a(n)/a(n-1) ~ log(a(n)) ~ prime(n). - _Thomas Ordowski_, Mar 26 2015
%F 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
%F Sum_{n>=1} 1/a(n) = A292667. - _Amiram Eldar_, Oct 15 2020
%p seq((ithprime@@n)(1),n=0..10); # _Peter Luschny_, Oct 16 2012
%t NestList[Prime@# &, 1, 16] (* _Robert G. Wilson v_, May 30 2006 *)
%o (PARI) print1(p=1);until(,print1(","p=prime(p))) \\ _M. F. Hasler_, Oct 09 2011
%o (Haskell)
%o a007097 n = a007097_list !! n
%o a007097_list = iterate a000040 1 -- _Reinhard Zumkeller_, Jul 14 2013
%o (GAP) P:=Filtered([1..60000],IsPrime);;
%o a:=[1];; for n in [2..10] do a[n]:=P[a[n-1]]; od; a; # _Muniru A Asiru_, Dec 22 2018
%Y Row 1 of array A114537.
%Y Left edge of tree A227413, right edge of A246378.
%Y Cf. A000040, A000720, A049076-A049081, A049084, A057450, A109301, A131842, A006450, A292667.
%Y Cf. A078442, A109082 (left inverses).
%Y Subsequence of A245823.
%K nonn,hard,nice
%O 0,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E a(15) corrected and a(16)-a(17) added by _Paul Zimmermann_
%E a(18)-a(19) found by _David Baugh_ using a program by Xavier Gourdon and _Andrey V. Kulsha_, Oct 25 2007
%E a(20)-a(21) found by _Andrey V. Kulsha_ using a program by Xavier Gourdon, Oct 02 2011
%E a(22) from _Henri Lifchitz_, Oct 14 2014
%E a(23) from _David Baugh_ using Kim Walisch's primecount, May 16 2016
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