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

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

Author

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

Keywords

• nonn
• hard
• nice

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• Lubomir Alexandrov, unpublished notes, circa 1960.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

• 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

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.

Programs

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

Haskell

a007097 n = a007097_list !! n
a007097_list = iterate a000040 1  -- Reinhard Zumkeller, Jul 14 2013

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 *)

PARI

print1(p=1);until(,print1(","p=prime(p)))  \\ M. F. Hasler, Oct 09 2011

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

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