A049076 -id:A049076 - cp's OEIS frontend

A049202 Primes p whose order of primeness A049076(p) is >= 6.

127, 709, 5381, 15299, 52711, 87803, 167449, 219613, 318211, 506683, 648391, 919913, 1128889, 1254739, 1471343, 1828669, 2269733, 2364361, 3042161, 3338989, 3509299, 4030889, 4535189, 5054303, 5823667, 6478961, 6816631

Offset: 1

Neil Fernandez

nonn

Union of A058322, A058324-A058328, A093046 etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert G. Wilson v)
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.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A000040, A006450, A038580, A049090, A049203, A057849, A057850, A057851, A057847, A058332, A093047.

map(ithprime@@4,select(isprime, [$1..137])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[35], 6] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

More terms from Robert G. Wilson v, Nov 10 2000

Name corrected by Sean A. Irvine, Jul 21 2021

A058325 Primes for which A049076(p) = 9.

Original entry on oeis.org

5381, 2269733, 17624813, 50728129, 77557187, 131807699, 259336153, 368345293, 440817757, 563167303, 751783477, 1107276647, 1170710369, 1367161723, 1760768239, 2062666783, 2323114841, 2458721501, 2621760397, 2860139341

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058326, A058327, A058328, A093046, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 8] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058324(n)). - R. J. Mathar, Jul 07 2012

A058326 Primes for which A049076(p) = 10.

Original entry on oeis.org

52711, 37139213, 326851121, 997525853, 1559861749, 2724711961, 5545806481, 8012791231, 9672485827, 12501968177, 16917026909, 25366202179, 26887732891, 31621854169, 41192432219, 48596930311, 55022031709, 58379844161

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

N. Fernandez, An order of primeness, F(p)N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058327, A058328, A093046, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 9] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058325(n)). - R. J. Mathar, Jul 07 2012

A058327 Primes for which A049076(p) = 11.

Original entry on oeis.org

648391, 718064159, 7069067389, 22742734291, 36294260117, 64988430769, 136395369829, 200147986693, 243504973489, 318083817907, 435748987787, 664090238153, 705555301183, 835122557939, 1099216100167, 1305164025929

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058328, A093046, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 10] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058326(n)). - R. J. Mathar, Jul 07 2012

A283458 Primes for which A049076(p) = 14.

Original entry on oeis.org

3657500101, 12055296811267, 156740126985437, 575411103069067, 966399998477597, 1841803943951113, 4176603711876241, 6373890505436101, 7910004791442043, 10613343313176589, 15000987504638299, 23825707567607467, 25462803625208449, 30634679101122821, 41400950264534519, 49969246522326097

Offset: 1

Robert G. Wilson v, Mar 08 2017

nonn

Also used Kim Walisch's primecount.

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A006450, A114537, A007821, A049078, A049079, A049080, A049081, A058322, A058324,Cf. A058325, A058326, A058327, A058328, A093046, A283459.

Nest[Prime, Select[Range[7], ! PrimeQ[#] &], 13]

a(n) = A000040(A093046(n)).

A283459 Primes for which A049076(p) = 15.

Original entry on oeis.org

88362852307, 392654585611999, 5519908106212193, 21034688742654437, 35843152090509943, 69532764058102673, 161191749822468689, 248761474969923757, 310467261969020581, 419776921940182991, 598644471430113247, 962125183414225879, 1029970322316321083, 1244984735583648473, 1695313841631390713

Offset: 1

Robert G. Wilson v, Mar 08 2017

nonn

Also used Kim Walisch's primecount.

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A006450, A114537, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A283458.

Nest[Prime, Select[Range[3], ! PrimeQ[#] &], 14]

a(n) = A000040(A283458(n)).

A081877 Duplicate of A049076.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1

Offset: 1

dead

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

Original entry on oeis.org

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

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

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

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).

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

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.

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

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

seq((ithprime@@n)(1),n=0..10); # Peter Luschny, Oct 16 2012

NestList[Prime@# &, 1, 16] (* Robert G. Wilson v, May 30 2006 *)

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

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

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

A006450 Prime-indexed primes: primes with prime subscripts.

Original entry on oeis.org

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, 1031, 1063, 1087, 1153, 1171, 1201, 1217, 1297, 1409, 1433, 1447, 1471

Offset: 1

Jeffrey Shallit, Nov 25 1975

A000040 = A006450 U A007821. - Juri-Stepan Gerasimov, Sep 24 2009
Subsequence of A175247 (primes (A000040) with noncomposite (A008578) subscripts), a(n) = A175247(n+1). - Jaroslav Krizek, Mar 13 2010
Primes p such that p and pi(p) are both primes. - Juri-Stepan Gerasimov, Jul 14 2011
Sum_{n>=1} 1/a(n) converges. In fact, Sum_{n>N} 1/a(n) < 1/log(N), by the integral test. - Jonathan Sondow, Jul 11 2012
The number of such primes not exceeding x > 0 is pi(pi(x)). I conjecture that the sequence a(n)^(1/n) (n = 1,2,3,...) is strictly decreasing. This is an analog of the Firoozbakht conjecture on primes. - Zhi-Wei Sun, Aug 17 2015
Limit_{n->infinity} a(n)/(n*(log(n))^2) = 1. Proof: By Cipolla's asymptotic formula, prime(n) ~ L(n) + R(n), where L(n)/n = log(n) + log(log(n)) - 1 and R(n)/n decreases logarithmically to 0. Hence, for large n, a(n) = prime(prime(n)) ~ L(L(n)+R(n)) + R(L(n)+R(n)) = n*(log(n))^2 + r(n), where r(n) grows as O(n*log(n)*log(log(n))). The rest of the proof is trivial. The convergence is very slow: for k = 1,2,3,4,5,6, sqrt(a(10^k)/10^k)/log(10^k) evaluates to 2.055, 1.844, 1.695, 1.611, 1.545, and 1.493, respectively. - Stanislav Sykora, Dec 09 2015

			a(5) = 31 because a(5) = p(p(5)) = p(11) = 31.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. S. Kimberley, Table of n, a(n) for n = 1..100000
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.
Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.
K. A. Broughan and A. R. Barnett, On the subsequence of primes having prime subscripts, JIS 12 (2009) 09.2.3.
Paul Cooijmans, Numbers.
Paul Cooijmans, Short Test For Genius.
R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM 22 (1975) 380-381.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. Fernandez, More terms of this and other sequences related to A049076.
A. B. Frizell, The permutations of the natural numbers can not be well ordered, Bull. Amer. Math. Soc. 22 (1915), no. 2, 71-73.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
J. Shallit, Letter to N. J. A. Sloane, Oct. 1975
Eric Weisstein's World of Mathematics, Prime formulas, see Cipolla formula.

Primes for which A049076 > 1.
Cf. A000040, A007821, A038580, A049090, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.
Cf. A185723 and A214296 for numbers and primes that are sums of distinct a(n); cf. A213356 and A185724 for those that are not.
Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270795, A270796, A102616.

a006450 = a000040 . a000040
a006450_list = map a000040 a000040_list
-- Reinhard Zumkeller, Jan 12 2013

[ NthPrime(NthPrime(n)): n in [1..51] ]; // Jason Kimberley, Apr 02 2010

seq(ithprime(ithprime(i)),i=1..50); # Uli Baum (Uli_Baum(AT)gmx.de), Sep 05 2007
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

Mathematica
```
Table[ Prime[ Prime[ n ] ], {n, 100} ]
```

i=0;forprime(p=2,1e4,if(isprime(i++),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

a=vector(10^3,n,prime(prime(n))) \\ Stanislav Sykora, Dec 09 2015

from sympy import prime
def a(n): return prime(prime(n))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Aug 11 2021

# much faster version for initial segment of sequence
from sympy import nextprime, isprime
def aupton(terms):
    alst, p, pi = [], 2, 1
    while len(alst) < terms:
        if isprime(pi): alst.append(p)
        p, pi = nextprime(p), pi+1
    return alst
print(aupton(10000)) # Michael S. Branicky, Aug 11 2021

a(n) = prime(prime(n)) = A000040(A000040(n)). - Juri-Stepan Gerasimov, Sep 24 2009
a(n) > n*(log(n))^2, as prime(n) > n*log(n) by Rosser's theorem. - Jonathan Sondow, Jul 11 2012
a(n)/log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 30 2015
Sum_{n>=1} 1/a(n) is in the interval (1.04299, 1.04365) (Bayless et al., 2013). - Amiram Eldar, Oct 15 2020

A061775 Number of nodes in rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7, 6, 6, 7, 6, 6, 7, 6, 7, 7, 6, 6, 7, 7, 6, 7, 6, 7, 8, 7, 7, 7, 7, 8, 7, 7, 6, 8, 8, 7, 7, 7, 6, 8, 7, 7, 8, 7, 8, 8, 6, 7, 8, 8, 7, 8, 7, 7, 9, 7, 8, 8, 7, 8, 9, 7, 7, 8, 8, 7, 8, 8, 7, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 7, 8, 8, 8, 9, 7, 7, 9

Offset: 1

N. J. A. Sloane, Jun 22 2001

nonn

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration).

Each n occurs A000081(n) times.

			a(4) = 3 because the rooted tree corresponding to the Matula-Goebel number 4 is "V", which has one root-node and two leaf-nodes, three in total.
See also the illustrations in A061773.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Keith Briggs, Matula numbers and rooted trees
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

One more than A196050.
Sum of entries in row n of irregular table A214573.
Number of entries in row n of irregular tables A182907, A206491, A206495 and A212620.
One less than the number of entries in row n of irregular tables A184187, A193401 and A193403.
Cf. A005517 (the position of the first occurrence of n).
Cf. A005518 (the position of the last occurrence of n).
Cf. A091233 (their difference plus one).
Cf. A214572 (Numbers k such that a(k) = 8).
Cf. also A000081, A061773, A049084, A020639, A049076, A078442, A091238, A091204, A091205, A109082, A127301, A109129, A193402, A193405, A193406, A196047, A196068, A198333, A206487, A206494, A206496, A214569, A214571, A213670, A214568, A228599, A245817, A245818.
Cf. also A075166, A075167, A216648, A216649.

import Data.List (genericIndex)
a061775 n = genericIndex a061775_list (n - 1)
a061775_list = 1 : g 2 where
   g x = y : g (x + 1) where
      y = if t > 0 then a061775 t + 1 else a061775 u + a061775 v - 1
          where t = a049084 x; u = a020639 x; v = x `div` u
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local u, v: u := n-> op(1, factorset(n)): v := n-> n/u(n): if n = 1 then 1 elif isprime(n) then 1+a(pi(n)) else a(u(n))+a(v(n))-1 end if end proc: seq(a(n), n = 1..108); # Emeric Deutsch, Sep 19 2011

a[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1+a[PrimePi[n]], a[u[n]]+a[v[n]]-1]]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jan 16 2014, after Emeric Deutsch *)

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
for(n=1, 10000, write("b061775.txt", n, " ", A061775(n)));
\\ Antti Karttunen, Aug 16 2014

from functools import lru_cache
from sympy import isprime, factorint, primepi
@lru_cache(maxsize=None)
def A061775(n):
    if n == 1: return 1
    if isprime(n): return 1+A061775(primepi(n))
    return 1+sum(e*(A061775(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1) = 1; if n = p_t (= the t-th prime), then a(n) = 1+a(t); if n = uv (u,v>=2), then a(n) = a(u)+a(v)-1.
a(n) = A091238(A091204(n)). - Antti Karttunen, Jan 2004
a(n) = A196050(n)+1. - Antti Karttunen, Aug 16 2014

More terms from David W. Wilson, Jun 25 2001

Extended by Emeric Deutsch, Sep 19 2011

cp's OEIS Frontend

A049202 Primes p whose order of primeness A049076(p) is >= 6.

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A058325 Primes for which A049076(p) = 9.

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A058326 Primes for which A049076(p) = 10.

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A058327 Primes for which A049076(p) = 11.

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A283458 Primes for which A049076(p) = 14.

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A283459 Primes for which A049076(p) = 15.

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A081877 Duplicate of A049076.

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

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