A093046 -id:A093046 - cp's OEIS frontend

A007821 Primes p such that pi(p) is not prime.

2, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 181, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 313, 317, 337, 347, 349, 359, 373

Offset: 1

Monte J. Zerger (mzerger(AT)cc4.adams.edu), Clark Kimberling

nonn

Primes prime(k) such that A049076(k)=1, sorted along increasing k. - R. J. Mathar, Jan 28 2014

The complement of A006450 (primes with prime index) within the primes A000040.

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

R. Zumkeller, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

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

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.

a007821 = a000040 . a018252
a007821_list = map a000040 a018252_list
-- Reinhard Zumkeller, Jan 12 2013

A007821 := proc(n) ithprime(A018252(n)) ; end proc: # R. J. Mathar, Jul 07 2012

Prime[ Select[ Range[75], !PrimeQ[ # ] &]] (* Robert G. Wilson v, Mar 15 2004 *)
With[{nn=100},Pick[Prime[Range[nn]],Table[If[PrimeQ[n],0,1],{n,nn}],1]] (* Harvey P. Dale, Aug 14 2020 *)

forprime(p=2, 1e3, if(!isprime(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

from sympy import primepi
def A007821(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-(p:=primepi(x))+primepi(p)
    return bisection(f,n,n) # Chai Wah Wu, Oct 19 2024

A137588(a(n)) = n; a(n) = A000040(A018252(n)). - Reinhard Zumkeller, Jan 28 2008
A000040 = A007821 U A006450. - Juri-Stepan Gerasimov, Sep 24 2009
A175247 U { a(n); n > 1 } = A000040. { a(n) } = { 2 } U { primes (A000040) with composite index (A002808) }. - Jaroslav Krizek, Mar 13 2010
G.f. over nonprime powers: Sum_{k >= 1} prime(k)*x^k-prime(prime(k))*x^prime(k). - Benedict W. J. Irwin, Jun 11 2016

Edited by M. F. Hasler, Jul 31 2015

A114537 Dispersion of the primes (an array read by downward antidiagonals).

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 17, 13, 8, 11, 59, 41, 19, 9, 31, 277, 179, 67, 23, 10, 127, 1787, 1063, 331, 83, 29, 12, 709, 15299, 8527, 2221, 431, 109, 37, 14, 5381, 167449, 87803, 19577, 3001, 599, 157, 43, 15, 52711, 2269733, 1128889, 219613, 27457, 4397, 919, 191, 47

Offset: 1

Clark Kimberling, Dec 07 2005

A number is prime if and only if it does not lie in Column 1. As a sequence, a permutation of the natural numbers. The fractal sequence of this dispersion is A022447 and the transposition sequence is A114538.

The dispersion of the composite numbers is given at A114577.

			Northwest corner of the Primeness array:
1   2   3    5    11     31     127       709       5381       52711        648391
4   7  17   59   277   1787   15299    167449    2269733    37139213     718064159
6  13  41  179  1063   8527   87803   1128889   17624813   326851121    7069067389
8  19  67  331  2221  19577  219613   3042161   50728129   997525853   22742734291
9  23  83  431  3001  27457  318211   4535189   77557187  1559861749   36294260117
10  29 109  599  4397  42043  506683   7474967  131807699  2724711961   64988430769
12  37 157  919  7193  72727  919913  14161729  259336153  5545806481  136395369829
14  43 191 1153  9319  96797 1254739  19734581  368345293  8012791231  200147986693
15  47 211 1297 10631 112129 1471343  23391799  440817757  9672485827  243504973489
16  53 241 1523 12763 137077 1828669  29499439  563167303 12501968177  318083817907
18  61 283 1847 15823 173867 2364361  38790341  751783477 16917026909  435748987787
20  71 353 2381 21179 239489 3338989  56011909 1107276647 25366202179  664090238153
21  73 367 2477 22093 250751 3509299  59053067 1170710369 26887732891  705555301183
22  79 401 2749 24859 285191 4030889  68425619 1367161723 31621854169  835122557939
24  89 461 3259 30133 352007 5054303  87019979 1760768239 41192432219 1099216100167
25  97 509 3637 33967 401519 5823667 101146501 2062666783 48596930311 1305164025929
26 101 547 3943 37217 443419 6478961 113256643 2323114841 55022031709 1484830174901
27 103 563 4091 38833 464939 6816631 119535373 2458721501 58379844161 1579041544637

Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)
Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria, 45 (1997) 157-168.

Robert G. Wilson v, Table of n, a(n) for n = 1..172
Neil Fernandez, An order of primeness, F(p)
Neil Fernandez, An order of primeness [cached copy, included with permission of the author]
Neil Fernandez, The Exploring Primeness Project
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Robert G. Wilson v, The Northwest Corner of the Primeness Array (24 x 24).

Columns 1-13: A018252, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046.
Rows 1-7: A007097, A057450, A057451, A057452, A057453, A057456, A057457.
Diagonal: A181441.
Cf. A000040, A007821, A114538, A006450.
If the antidiagonals are read in the opposite direction we get A138947.

A114537 := proc(r,c) option remember; if c = 1 then A018252(r) ; else ithprime(procname(r,c-1)) ; end if; end proc: # R. J. Mathar, Oct 22 2010

NonPrime[n_] := FixedPoint[n + PrimePi@# + 1 &, n]; t[n_, k_] := Nest[Prime, NonPrime[n], k]; Table[ t[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten
(* or to view the table *) Table[t[n, k], {n, 0, 6}, {k, 0, 10}] // TableForm (* Robert G. Wilson v, Dec 26 2005 *)

T(r,1) = A018252(r). T(r,c) = prime(T(r,c-1)), c>1. [R. J. Mathar, Oct 22 2010]

A049078 Primes prime(k) for which A049076(k) = 2.

Original entry on oeis.org

3, 17, 41, 67, 83, 109, 157, 191, 211, 241, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1087, 1171, 1201, 1217, 1409, 1433, 1447, 1471, 1499, 1597, 1621, 1669, 1723, 1741, 1823, 1913, 2027, 2063, 2081, 2099

Offset: 1

Neil Fernandez

nonn

			For these primes S(p) is a prime but S(S(p)) is not. E.g. S(17)=7, S(7)=4.

Charles R Greathouse IV, 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. A049076, A007821, A049079-A049081, A058322, A058324-A058328, A093046, A006450, A018252, A236542.

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.

A049078 := proc(n) ithprime(A007821(n)) ; end proc: # R. J. Mathar, Aug 14 2008

Nest[ Prime, Select[ Range[70], !PrimeQ[ # ] &], 2] (* Robert G. Wilson v, Mar 15 2004 *)

is(n)=my(p=primepi(n)); isprime(p) && !isprime(primepi(p)) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = prime(A007821(n)). - Juri-Stepan Gerasimov, Aug 11 2008

a(n) ~ A006450(n) ~ n log^2 n. - Charles R Greathouse IV, Apr 29 2015

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A049081 Primes prime(k) for which A049076(k) = 5.

Original entry on oeis.org

31, 1787, 8527, 19577, 27457, 42043, 72727, 96797, 112129, 137077, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, 718807, 755387, 839483, 864013, 985151, 1021271, 1080923, 1159901, 1278779

Offset: 1

Neil Fernandez

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
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, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049081 := proc(n)
        ithprime(A049080(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[44], !PrimeQ[ # ] &], 5] (* Robert G. Wilson v, Mar 15 2004 *)

More terms from Robert G. Wilson v, Dec 12 2000

A049079 Primes prime(k) for which A049076(k) = 3.

Original entry on oeis.org

5, 59, 179, 331, 431, 599, 919, 1153, 1297, 1523, 1847, 2381, 2477, 2749, 3259, 3637, 3943, 4091, 4273, 4549, 5623, 5869, 6113, 6661, 6823, 7607, 7841, 8221, 8719, 9461, 9739, 9859, 11743, 11953, 12097, 12301, 12547, 13469, 13709, 14177, 14723, 14867

Offset: 1

Neil Fernandez

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, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049079 := proc(n)
        ithprime(A049078(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[60], !PrimeQ[ # ] &], 3] (* Robert G. Wilson v, Mar 15 2004 *)

Definition edited by Zak Seidov, Sep 15 2013

A058322 Primes for which A049076(p) = 7.

Original entry on oeis.org

127, 15299, 87803, 219613, 318211, 506683, 919913, 1254739, 1471343, 1828669, 2364361, 3338989, 3509299, 4030889, 5054303, 5823667, 6478961, 6816631, 7220981, 7807321, 10311439, 10875143, 11469013, 12838937, 13243033, 15239333, 15837299, 16827557, 18143603

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, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A058322 := proc(n)
        ithprime(A049081(n)) ;
end proc: # R. J. Mathar, Jul 07 2012
# second Maple program:
map(ithprime@@6, remove(isprime, [$1..42]))[];  # Alois P. Heinz, Mar 15 2020

Nest[ Prime, Select[ Range[40], !PrimeQ[ # ] &], 6] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A049081(n)).

More terms from Alois P. Heinz, Mar 15 2020

A049080 Primes prime(k) for which A049076(k) = 4.

Original entry on oeis.org

11, 277, 1063, 2221, 3001, 4397, 7193, 9319, 10631, 12763, 15823, 21179, 22093, 24859, 30133, 33967, 37217, 38833, 40819, 43651, 55351, 57943, 60647, 66851, 68639, 77431, 80071, 84347, 90023, 98519, 101701, 103069, 125113, 127643

Offset: 1

Neil Fernandez

nonn

Jens Kruse Andersen, 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. A049076, A007821, A049078, A049079, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049080 := proc(n)
        ithprime(A049079(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[50], !PrimeQ[ # ] &], 4] (* Robert G. Wilson v, Mar 15 2004 *)

A049203 Primes for which A049076(p) >= 5.

Original entry on oeis.org

31, 127, 709, 1787, 5381, 8527, 15299, 19577, 27457, 42043, 52711, 72727, 87803, 96797, 112129, 137077, 167449, 173867, 219613, 239489, 250751, 285191, 318211, 352007, 401519, 443419, 464939, 490643, 506683, 527623, 648391, 683873, 718807

Offset: 1

Neil Fernandez

nonn

Union of A049081, A058322, A058324-A058328, A093046, etc. - R. J. Mathar, Jul 07 2012

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

Cf. A000040, A006450, A038580, A049076, A049090, A049202, A057847, A057849, A057850, A057851, A058332, A093047.

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

Nest[ Prime, Range[35], 5] (* Robert G. Wilson v, Mar 15 2004 *)

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

a(n) = A000040(A049090(n)). - R. J. Mathar, Jul 07 2012

a(n) ~ n (log n)^5. - 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

A058324 Primes for which A049076(p) = 8.

Original entry on oeis.org

709, 167449, 1128889, 3042161, 4535189, 7474967, 14161729, 19734581, 23391799, 29499439, 38790341, 56011909, 59053067, 68425619, 87019979, 101146501, 113256643, 119535373, 127065427, 138034009, 185350441, 196100297, 207460717, 233784751, 241568891, 280256489

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, A058325, A058326, A058327, A058328, A093046, A006450.

map(ithprime@@7, remove(isprime, [$1..38]))[];  # Alois P. Heinz, Mar 15 2020

Nest[ Prime, Select[ Range[34], !PrimeQ[ # ] &], 7] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058322(n)). - R. J. Mathar, Jul 07 2012

More terms from Alois P. Heinz, Mar 15 2020

A058328 Primes for which A049076(p) = 12.

Original entry on oeis.org

9737333, 16123689073, 175650481151, 592821132889, 963726515729, 1765037224331, 3809491708961, 5669795882633, 6947574946087, 9163611272327, 12695664159413, 19638537755027, 20909033866927, 24894639811901

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, A058327, A093046, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 11] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058327(n)). - R. J. Mathar, Jul 07 2012

cp's OEIS Frontend

A007821 Primes p such that pi(p) is not prime.

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A114537 Dispersion of the primes (an array read by downward antidiagonals).

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A049078 Primes prime(k) for which A049076(k) = 2.

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A049081 Primes prime(k) for which A049076(k) = 5.

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A049079 Primes prime(k) for which A049076(k) = 3.

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A058322 Primes for which A049076(p) = 7.

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A049080 Primes prime(k) for which A049076(k) = 4.

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