A078442 -id:A078442 - cp's OEIS frontend

A057847 Primes p whose order of primeness A078442(p) is at least 10.

648391, 9737333, 174440041, 718064159, 3657500101, 7069067389, 16123689073, 22742734291, 36294260117, 64988430769, 88362852307, 136395369829, 175650481151, 200147986693, 243504973489, 318083817907, 414507281407

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 221 terms from Robert G. Wilson v)
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A058332, A093047.

Nest[ Prime, Range[35], 10] (* Robert G. Wilson v, Mar 15 2004 *)

a(n)=my(k=11);while(k--,n=prime(n));n \\ Charles R Greathouse IV, Feb 24 2017

a(n) = prime(A057851(n)). - Andrew Howroyd, Nov 17 2024

Name clarified by Andrew Howroyd, Nov 17 2024

A057849 Primes p whose order of primeness A078442(p) is at least 7.

Original entry on oeis.org

709, 5381, 52711, 167449, 648391, 1128889, 2269733, 3042161, 4535189, 7474967, 9737333, 14161729, 17624813, 19734581, 23391799, 29499439, 37139213, 38790341, 50728129, 56011909, 59053067, 68425619, 77557187, 87019979, 101146501, 113256643, 119535373, 127065427

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..1000 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. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057850, A057851, A057847, A058332, A093047.

a:= ithprime@@7;
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 14 2015

Nest[ Prime, Range[35], 7] (* Robert G. Wilson v, Mar 15 2004 *)

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

Name clarified by Andrew Howroyd, Nov 17 2024

A057850 Primes p whose order of primeness A078442(p) is at least 8.

Original entry on oeis.org

5381, 52711, 648391, 2269733, 9737333, 17624813, 37139213, 50728129, 77557187, 131807699, 174440041, 259336153, 326851121, 368345293, 440817757, 563167303, 718064159, 751783477, 997525853, 1107276647, 1170710369, 1367161723

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Union of A058325-A058328, A093046 etc. - R. J. Mathar, Jul 07 2012

Robert G. Wilson v, Table of n, a(n) for n = 1..11457
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. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057851, A057847, A058332, A093047.

Nest[ Prime, Range[35], 8] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u, vv, w); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++) && isprime(vv++) && isprime(w++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = A049090(A049090(n)). - James G. Merickel, Feb 14 2010

a(n) = A000040(A057849(n)). - R. J. Mathar, Jul 07 2012

Name clarified by Andrew Howroyd, Nov 17 2024

A057851 Primes p whose order of primeness A078442(p) is at least 9.

Original entry on oeis.org

52711, 648391, 9737333, 37139213, 174440041, 326851121, 718064159, 997525853, 1559861749, 2724711961, 3657500101, 5545806481, 7069067389, 8012791231, 9672485827, 12501968177, 16123689073, 16917026909, 22742734291

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1381 terms from Robert G. Wilson v)
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057847, A058332, A093047.

Nest[ Prime, Range[35], 9] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u, vv, w, x); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++) && isprime(vv++) && isprime(w++) && isprime(x++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 19 2017

Name clarified by Andrew Howroyd, Nov 17 2024

A058332 Primes p whose order of primeness A078442(p) is at least 11.

Original entry on oeis.org

9737333, 174440041, 3657500101, 16123689073, 88362852307, 175650481151, 414507281407, 592821132889, 963726515729, 1765037224331, 2428095424619, 3809491708961, 4952019383323, 5669795882633, 6947574946087, 9163611272327

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..47.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A093047.

Nest[ Prime, Range[35], 11] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = prime(A057847(n)). - Andrew Howroyd, Nov 17 2024

Name clarified by Andrew Howroyd, Nov 17 2024

A093047 Primes p whose order of primeness A078442(p) is at least 12.

Original entry on oeis.org

174440041, 3657500101, 88362852307, 414507281407, 2428095424619, 4952019383323, 12055296811267, 17461204521323, 28871271685163, 53982894593057, 75063692618249, 119543903707171, 156740126985437, 180252380737439, 222334565193649

Offset: 1

Robert G. Wilson v, Mar 15 2000

Primes p whose primeness is > 12: 3657500101, 88362852307, 2428095424619, 12055296811267, 75063692618249, 156740126985437, ..., . - Robert G. Wilson v, Mar 15 2000

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

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A058332, A093047.

Nest[ Prime, Range[35], 12] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A058332(prime(n)). - Andrew Howroyd, Nov 17 2024

Name clarified by Andrew Howroyd, Nov 17 2024

A333353 Primes p whose order of primeness A078442(p) is prime.

Original entry on oeis.org

3, 5, 17, 31, 41, 59, 67, 83, 109, 157, 179, 191, 211, 241, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, 1031, 1087, 1153, 1171, 1201, 1217, 1297, 1409, 1433, 1447, 1471, 1499, 1523, 1597, 1621

Offset: 1

Alois P. Heinz, Mar 15 2020

nonn

			31 is a term: 31 -> 11 -> 5 -> 3 -> 2 -> 1, five (a prime number of) steps "->" = pi = A000720.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A000040, A000720, A049076, A078442, A236542, A262275, A333364.

b:= proc(n) option remember;
      `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if isprime(b(p)) then break fi
      od; p
    end:
seq(a(n), n=1..55);

b[n_] := b[n] = If[!PrimeQ[n], 0, 1+b[PrimePi[n]]];
okQ[n_] := PrimeQ[n] && PrimeQ[b[n]];
Select[Range[2000], okQ] (* Jean-François Alcover, May 30 2022 *)

{ p in primes : A078442(p) is prime }.

a(n) = prime(A333364(n)).

A333364 Indices of primes p whose order of primeness A078442(p) is prime.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Alois P. Heinz, Mar 16 2020

nonn

All terms are prime.

			11 is a term: prime(11) = 31 -> 11 -> 5 -> 3 -> 2 -> 1, five (a prime number of) steps "->" = pi = A000720.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A000040, A000720, A049076, A333353.

b:= proc(n) option remember;
      `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if isprime(b(p)+1) then break fi
      od; p
    end:
seq(a(n), n=1..62);

b[n_] := b[n] = If[PrimeQ[n], 1 + b[PrimePi[n]], 0];
a[n_] := a[n] = Module[{p}, p = If[n == 1, 1, a[n - 1]];
   While[True, p = NextPrime[p]; If[PrimeQ[b[p] + 1], Break[]]]; p];
Table[a[n], {n, 1, 62}] (* Jean-François Alcover, Sep 14 2022, after Alois P. Heinz *)

{ p in primes : A049076(p) is prime }.

a(n) = pi(A333353(n)), with pi = A000720.

A094722 Primes p whose order of primeness A078442(p) is at least 13.

Original entry on oeis.org

3657500101, 88362852307, 2428095424619, 12055296811267, 75063692618249, 156740126985437

Offset: 1

Robert G. Wilson v, May 22 2004

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

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A058332, A093047.

Mathematica
```
Nest[ Prime, Range[35], 13]
```

a(n) = prime(A093047(n)). - Andrew Howroyd, Nov 17 2024

a(6) from Robert G. Wilson v, May 22 2004

Name clarified by Andrew Howroyd, Nov 17 2024

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

cp's OEIS Frontend

A057847 Primes p whose order of primeness A078442(p) is at least 10.

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A057849 Primes p whose order of primeness A078442(p) is at least 7.

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A057850 Primes p whose order of primeness A078442(p) is at least 8.

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A057851 Primes p whose order of primeness A078442(p) is at least 9.

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A058332 Primes p whose order of primeness A078442(p) is at least 11.

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A093047 Primes p whose order of primeness A078442(p) is at least 12.

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A333353 Primes p whose order of primeness A078442(p) is prime.

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A333364 Indices of primes p whose order of primeness A078442(p) is prime.

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