

A007097


Primeth recurrence: a(n+1) = a(n)th prime.
(Formerly M0734)


281



1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041
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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
a(n) is the MatulaGoebel number of the rooted path tree on n+1 vertices. The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel 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


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Lubomir Alexandrov, unpublished notes, circa 1960.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, "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 2429, 2000, Dubna, Russia, Book of Abstracts, p. 19. Available at arXiv:math/0105154 [math.NT], 2001.


FORMULA

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



MATHEMATICA



PROG

(PARI) print1(p=1); until(, print1(", "p=prime(p))) \\ M. F. Hasler, Oct 09 2011
(Haskell)
a007097 n = a007097_list !! n
(GAP) P:=Filtered([1..60000], IsPrime);;
a:=[1];; for n in [2..10] do a[n]:=P[a[n1]]; od; a; # Muniru A Asiru, Dec 22 2018


CROSSREFS



KEYWORD

nonn,hard,nice


AUTHOR



EXTENSIONS

a(20)a(21) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 02 2011
a(23) from David Baugh using Kim Walisch's primecount, May 16 2016


STATUS

approved



