A058322 -id:A058322 - cp's OEIS frontend

A093046 Primes for which A049076(p) = 13.

174440041, 414507281407, 4952019383323, 17461204521323, 28871271685163, 53982894593057, 119543903707171, 180252380737439, 222334565193649, 295872998567819, 414190707114539, 649544694886663, 692919372869953, 829484152743469, 1111923751842437, 1335294947809661, 1532021237514419, 1635795965187779

Offset: 1

Robert G. Wilson v, Mar 15 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, A058328, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 12]

a(n) = A000040(A058328(n)). - R. J. Mathar, Jul 07 2012

a(7)-a(9) from Robert G. Wilson v, Dec 27 2005

a(10)-a(18) from Robert G. Wilson v, Mar 08 2017 using Kim Walisch's primecount.

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

Original entry on oeis.org

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

A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

Original entry on oeis.org

1, 4, 9, 2, 16, 7, 6, 13, 3, 19, 26, 17, 8, 23, 41, 5, 12, 67, 10, 29, 59, 37, 14, 83, 179, 11, 43, 331, 20, 47, 39, 109, 277, 157, 53, 431, 22, 1063, 31, 191, 15, 2221, 27, 61, 211, 71, 30, 599, 1787, 919, 241, 3001, 35, 73, 8527, 127, 1153, 79, 21, 19577, 44, 89, 283

Offset: 1

Katarzyna Matylla, Feb 11 2008

nonn

Exchanges primes with composites, primeth primes with composith composites, etc.
Exchange the k-th prime of order j with the k-th composite of order j and vice versa.
Self-inverse permutation of positive integers.
If n is the composite number A236536(r,k), then a(n) is the corresponding prime A236542(r,k) at the same position (r,k). Vice versa, if n is the prime A236542(r,k), then a(n) is the corresponding composite A236536(r,k) at the same position. - Andrew Weimholt, Jan 28 2014
The original name for this entry did not produce this sequence, but instead A236854, which differs from this permutation for the first time at n=8, where A236854(8)=23, while here a(8)=13. - Antti Karttunen, Feb 01 2014

			From _Andrew Weimholt_, Jan 29 2014: (Start)
More generally, takes the primes organized in an array according to the sieving process described in the Fernandez paper:
        Row[1](n) = 2, 7, 13, 19, 23, ...
        Row[2](n) = 3, 17, 41, 67, 83, ...
        Row[3](n) = 5, 59, 179, ...
        Row[4](n) = 11, 277, ...
        Lets call this  T_p (n, k)
Also take the composites organized in a similar manner, except we use "composite" numbered positions in our sieve:
        Row[1](n) = 4, 6, 8, 10, 14, 20, 22, ...
        Row[2](n) = 9, 12, 15, 18, 24, ...
        Row[3](n) = 16, 21, 25, ...
        Lets call this T_c (n, k)
If we now take the natural numbers and swap each number (except for 1) with the number which holds the same spot in the other array, then we get the sequence: 1, 4, 9, 2, 16, 7, 6, 13, with for example a(8) = 13 (13 holds the same position in the 'prime' table as 8 does in the 'composite' table). (End)

R. J. Mathar, Table of n, a(n) for n = 1..197
N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index to permutations of positive integers

Cf. A000040, A007097, A049076, A049078 - A049081, A058322, A058324 - A058328, A093046, A002808, A006508, A059981, A078442, A236854.

A135044 := proc(n)
    if n = 1 then
        1;
    elif isprime(n) then
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236542(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236542(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236536(r,c) ;
    else
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236536(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236536(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236542(r,c) ;
    end if;
end proc: # R. J. Mathar, Jan 28 2014

Composite[n_Integer] := Block[{k = n + PrimePi@n + 1}, While[k != n + PrimePi@k + 1, k++ ]; k]; Compositeness[n_] := Block[{c = 1, k = n}, While[ !(PrimeQ@k || k == 1), k = k - 1 - PrimePi@k; c++ ]; c]; Primeness[n_] := Block[{c = 1, k = n}, While[ PrimeQ@k, k = PrimePi@k; c++ ]; c];
ckj[k_, j_] := Select[ Table[Composite@n, {n, 10000}], Compositeness@# == j &][[k]]; pkj[k_, j_] := Select[ Table[Prime@n, {n, 3000}], Primeness@# == j &][[k]]; f[0]=0; f[1] = 1;
f[n_] := If[ PrimeQ@ n, pn = Primeness@n; ckj[ Position[ Select[ Table[ Prime@ i, {i, 150}], Primeness@ # == pn &], n][[1, 1]], pn], cn = Compositeness@n; pkj[ Position[ Select[ Table[ Composite@ i, {i, 500}], Compositeness@ # == cn &], n][[1, 1]], cn]]; Array[f, 64] (* Robert G. Wilson v *)

a(1)=1, a(A236536(r,k))=A236542(r,k), a(A236542(r,k))=A236536(r,k)

Edited, corrected and extended by Robert G. Wilson v, Feb 18 2008

Name corrected by Andrew Weimholt, Jan 29 2014

A059395 Smaller of safe prime twins: special safe primes (A005385) p such that the next prime is also the next safe prime and is p+12, i.e., occurs at the closest possible distance, 12.

Original entry on oeis.org

467, 1307, 2447, 5087, 5927, 12527, 18947, 44687, 78467, 83207, 118787, 143687, 164987, 196907, 204587, 207227, 208787, 229487, 236507, 257627, 275987, 297707, 330887, 339827, 367007, 369647, 394007, 454907, 458807, 474347, 534827, 536087

Offset: 1

Labos Elemer, Jan 29 2001

nonn

			The pairs (5,7) and (7,11) are omitted, albeit are both consecutive primes and consecutive safe primes, however their distances (2 and 4) are singular. Cases [467, 439] and [20738027, 20738039] are pairs are both consecutive of consecutive primes and consecutive safe primes in minimal distance=12. The corresponding twins of Sophie Germain primes are [233, 239] or [1369013, 1369019] in distance 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005385, A005384, A058322, A059323, A059327.

safeQ[p_] := PrimeQ[(p-1)/2]; seq={}; c=0; p1 = p2 = 11; q1 = safeQ[p1]; While[c < 30, p2 = NextPrime[p2]; q2 = safeQ[p2]; If[q1 && q2 && p2 == p1 + 12, c++; AppendTo[seq, p1]]; p1 = p2; q1 = q2]; seq (* Amiram Eldar, Jan 13 2020 *)

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)).

cp's OEIS Frontend

A093046 Primes for which A049076(p) = 13.

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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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A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

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A059395 Smaller of safe prime twins: special safe primes (A005385) p such that the next prime is also the next safe prime and is p+12, i.e., occurs at the closest possible distance, 12.

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A283458 Primes for which A049076(p) = 14.

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