A007097 Primeth recurrence: a(n+1) = a(n)-th prime. (Formerly M0734)

1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041

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

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Lubomir Alexandrov, unpublished notes, circa 1960.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

Table of n, a(n) for n=0..23.

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]

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992

Index entries for sequences related to Matula-Goebel numbers

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

Sum_{n>=1} 1/a(n) = A292667. - Amiram Eldar, Oct 15 2020

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, A292667.

Cf. A078442, A109082 (left inverses).

Subsequence of A245823.

Sequence in context: A357220 A112279 A130166 * A173422 A132745 A124538

Adjacent sequences: A007094 A007095 A007096 * A007098 A007099 A007100

Keyword

nonn,hard,nice

Author

N. J. A. Sloane, Robert G. Wilson v

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