A049079 -id:A049079 - cp's OEIS frontend

A058325 Primes for which A049076(p) = 9.

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

A236542 Array T(n,k) read along descending antidiagonals: row n contains the primes with n steps in the prime index chain.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 28 2014

Row n contains the primes A000040(j) for which A049076(j) = n.

			The array starts:
    2,    7,   13,   19,   23,   29,   37,   43,   47,   53,...
    3,   17,   41,   67,   83,  109,  157,  191,  211,  241,...
    5,   59,  179,  331,  431,  599,  919, 1153, 1297, 1523,...
   11,  277, 1063, 2221, 3001, 4397, 7193, 9319,10631,12763,...
   31, 1787, 8527,19577,27457,42043,72727,96797,112129,137077,...

N. Fernandez, An order of primeness, F(p).

Cf. A007821 (row 1), A049078 (row 2), A049079 (row 3), A007097 (column 1), A058010 (diagonal), A057456 - A057457 (columns), A135044, A236536.

A236542 := proc(n,k)
    option remember ;
    if n = 1 then
        A007821(k) ;
    else
        ithprime(procname(n-1,k)) ;
    end if:
end proc:
for d from 2 to 10 do
    for k from d-1 to 1 by -1 do
            printf("%d,",A236542(d-k,k)) ;
    end do:
end do:

A007821 = Prime[Select[Range[15], !PrimeQ[#]&]];
T[n_, k_] := T[n, k] = If[n == 1, If[k <= Length[A007821], A007821[[k]], Print["A007821 must be extended"]; Abort[]], Prime[T[n-1, k]]];
Table[T[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 16 2020 *)

T(1,k) = A007821(k).

T(n,k) = prime( T(n-1,k) ), n>1 .

A050436 Third-order composites.

Original entry on oeis.org

16, 21, 25, 26, 28, 33, 36, 38, 39, 42, 48, 49, 50, 52, 55, 56, 57, 60, 64, 68, 69, 70, 72, 74, 77, 78, 80, 84, 87, 88, 90, 93, 94, 95, 98, 100, 104, 105, 106, 110, 111, 115, 117, 118, 119, 121, 122, 124, 125, 126, 130, 133, 135, 138, 140, 141, 145, 146, 147

Offset: 1

Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

			C(C(C(8))) = C(C(15)) = C(25) = 38. So 38 is in the sequence.

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

Cf. A049076, A049077, A049078, A049079, A049080, A049081, A006450, A050435, A050438, ...

C := remove(isprime,[$4..1000]): seq(C[C[C[C[n]]]],n=1..100);

Nest[Values@ KeySelect[MapIndexed[First@ #2 -> #1 &, #], CompositeQ] &, Select[Range@ 150, CompositeQ], 2] (* Michael De Vlieger, Jul 22 2017 *)

Let C(n) be the n-th composite number, with C(1)=4. Then these are numbers C(C(C(n))).

More terms from Asher Auel Dec 15 2000

A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.

Original entry on oeis.org

1, 2, 6, 13, 22, 79, 108, 593, 722, 5471, 6290, 62653, 69558, 876329, 951338, 14679751, 15692307, 289078661, 305618710, 6588286337, 6908033000, 171482959009, 178668550322, 5040266614919, 5225256019175, 165678678591359, 171068472492228, 6039923990345039

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..31
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064961, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1}; b = 1; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

from functools import cache
from sympy import prime, composite
@cache
def A064960(n): return 1 if n == 1 else composite(A064960(n-1)) if n % 2 else prime(A064960(n-1)) # Chai Wah Wu, Jan 01 2022

a(26)-a(28) from Chai Wah Wu, May 07 2018

A064961 Composite-then-prime recurrence; a(2n) = a(2n-1)-th composite and a(2n+1) = a(2n)-th prime and a(1) = 1.

Original entry on oeis.org

1, 4, 7, 14, 43, 62, 293, 366, 2473, 2892, 26317, 29522, 344249, 376259, 5429539, 5831545, 101291779, 107457490, 2198218819, 2310909505, 54720307351, 57128530327, 1543908890351, 1603146693999, 48871886538151, 50527531769529, 1720466016680911, 1772475453490311

Offset: 1

Robert G. Wilson v, Oct 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..32
Andrew R. Booker, The Nth Prime Page
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A007097, A006508 & A064960, see also A057450, A057451, A057452, A057453, A057456 & A057457 and A049076, A049077, A049078, A049079, A049080 & A049081.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; a = {1, 4}; b = 4; Do[ If[ !PrimeQ[b], b = Prime[b], b = Composite[b]]; a = Append[a, b], {n, 1, 23}]; a

a(24)-a(26) corrected and a(27)-a(28) added by Chai Wah Wu, Aug 22 2018

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)).

A318554 a(n) is the smallest prime number having order of primeness = prime(n).

Original entry on oeis.org

3, 5, 31, 709, 9737333, 3657500101, 2586559730396077, 4123221751654370051, 28785866289100396890228041

Offset: 1

David James Sycamore, Aug 27 2018

Let F(k) denote A049076(k). The list of primes p such that F(p) = n begins with q, the smallest prime to have prime index in each of n-1 successive primeth iterations, finally taking nonprime index 1 at the n-th iteration. All other members p such that F(p) = n are primes > q which also take a nonprime index at the n-th iteration. The reverse sequence of associated indices for q = prime(n) gives successive terms of the primeth recurrence 1,2,3,5,... until reaching A007097(prime(n)) = a(n).

			The sequence of primes with order of primeness F(p) = prime(1) = 2 begins 3,17,41,67,...
so a(1)=3. Likewise, F(p) = prime(2) = 3 begins 5,59,179,... so a(2)=5.

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

Cf. A000040, A006450, A007097, A049076, A049078, A049079, A049081.

a(n) = A007097(prime(n)); n >= 1.

cp's OEIS Frontend

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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A236542 Array T(n,k) read along descending antidiagonals: row n contains the primes with n steps in the prime index chain.

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A050436 Third-order composites.

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Maple

Mathematica

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Extensions

A064960 The prime then composite recurrence; a(2n) = a(2n-1)-th prime and a(2n+1) = a(2n)-th composite and a(1) = 1.

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A064961 Composite-then-prime recurrence; a(2n) = a(2n-1)-th composite and a(2n+1) = a(2n)-th prime and a(1) = 1.

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A283458 Primes for which A049076(p) = 14.

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