A023330 -id:A023330 - cp's OEIS frontend

A059456 Unsafe primes: primes not in A005385.

2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Labos Elemer, Feb 02 2001

nonn

A010051(a(n))*(1-A156659(a(n))) = 1; subsequence of A156657. - Reinhard Zumkeller, Feb 18 2009
Also, primes p such that p-1 is a non-semiprime. - Juri-Stepan Gerasimov, Apr 28 2010
Conjecture: From the sequence of prime numbers, let 2 and remove the first data iteration of 2*p+1; leave 3 and remove the prime data by the iteration 2*p+1 and we get the sequence. Example for p=2, remove(5,11,23,47); p=3, remove(7); p=13, p=17, p=19, p=23, remove(47); and so on. - Vincenzo Librandi, Aug 07 2010

			31 is here because (31-1)/2=15 is not prime. 2 and 3 are here because 1/2 and 1 are not prime numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330.

Initial terms for groups in A075712.

Complement[Prime@ Range@ PrimePi@ Max@ #, #] &@ Select[Prime@ Range@ 90, PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, May 01 2016 *)
Select[Prime[Range[100]],PrimeOmega[#-1]!=2&] (* Harvey P. Dale, May 13 2018 *)

is(n)=isprime(n) && !isprime(n\2) \\ Charles R Greathouse IV, May 02 2016

a(n) ~ n log n. - Charles R Greathouse IV, Dec 29 2024

A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

Original entry on oeis.org

3, 29, 53, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1049, 1103, 1223, 1289, 1499, 1559, 1583, 1601, 1733, 1973, 2003, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2543

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3} = {composite, prime, prime, composite}.

			53 is a term because 26 and 215 are composites, and 53 and 107 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ccftQ[p_]:=Boole[PrimeQ[{(p-1)/2,p,2 p+1,4 p+3}]]=={0,1,1,0}; Select[ Prime[ Range[400]],ccftQ] (* Harvey P. Dale, Jun 19 2021 *)

A059763 Primes starting a Cunningham chain of the first kind of length 4.

Original entry on oeis.org

509, 1229, 1409, 2699, 3539, 6449, 10589, 11549, 11909, 12119, 17159, 19709, 19889, 22349, 26189, 27479, 30389, 43649, 55229, 57839, 60149, 71399, 74699, 75329, 82499, 87539, 98369, 101399, 104369, 112919, 122099, 139439, 148829, 166739

Offset: 1

Labos Elemer, Feb 20 2001

Initial (unsafe) primes of Cunningham chains of first type with length exactly 4. Primes in A059453 that survive as primes just three "2p+1 iterations", forming chains of exactly 4 terms.
The definition indicates each chain is exactly 4 primes long (i.e., the chain cannot be a subchain of a longer one). That is why this sequence is different from A023272, which also gives primes included in longer chains ("starting" them or not).
Prime p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15} = {composite, prime, prime, prime, prime, composite}.

			1229 is a term because, through 2p+1, 1229 -> 2459 -> 4919 -> 9839 and the chain ends here since 2*9839 + 1 = 11*1789 is composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700.

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

isA059763 := proc(p) local pitr,itr ; if isprime(p) then if isprime( (p-1)/2 ) then RETURN(false) ; else pitr := p ; for itr from 1 to 3 do pitr := 2*pitr+1 ; if not isprime(pitr) then RETURN(false) ; fi ; od: pitr := 2*pitr+1 ; if isprime(pitr) then RETURN(false) ; else RETURN(true) ; fi ; fi ; else RETURN(false) ; fi ; end: for i from 2 to 100000 do p := ithprime(i) ; if isA059763(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Jul 23 2008

Edited and extended by R. J. Mathar, Jul 23 2008, Aug 18 2008

A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

Original entry on oeis.org

2, 53639, 53849, 61409, 66749, 143609, 167729, 186149, 206369, 268049, 296099, 340919, 422069, 446609, 539009, 594449, 607319, 658349, 671249, 725009, 775949, 812849, 819509, 926669, 1008209, 1092089, 1132949, 1271849

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15, 32p+31} = {nonprime, prime, prime, prime, prime, prime, composite}.

			2 is here because (2-1)/2 = 1/2 and 32*2+31 = 95 are not primes, while 2, 5, 11, 23, and 47 give a first-kind Cunningham chain of 5 primes which cannot be continued.
53639 is here because through <2p+1>, 53639 -> 107279 -> 214559 -> 429119 -> 858239 and the chain ends here (with this operator).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452-A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

l5Q[n_]:=Module[{a=PrimeQ[(n-1)/2],b=PrimeQ[ NestList[2#+1&,n,5]]}, Join[{a},b]=={False,True,True,True,True,True,False}]; Select[Range[ 1300000],l5Q] (* Harvey P. Dale, Oct 14 2012 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Entry revised by N. J. A. Sloane, Apr 01 2006

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

Original entry on oeis.org

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

			89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

lst={};Do[p=Prime[n];If[ !PrimeQ[(p-1)/2],If[PrimeQ[2*p+1],AppendTo[lst,p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)
Select[Prime[Range[300]],PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

is(p) = isprime(p) && isprime(2*p+1) && if(p > 2, !isprime((p-1)/2), 1); \\ Amiram Eldar, Jul 15 2024

from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*(1-A156659(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

Original entry on oeis.org

13, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 97, 101, 103, 109, 127, 137, 139, 149, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 257, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 439, 449

Offset: 1

Labos Elemer, Feb 05 2001

nonn

Primes which are neither safe nor of Sophie Germain type.
Primes not in Cunningham chains of the first kind. - Alonso del Arte, Jun 30 2005
A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

			Prime p=17 is here because both 35 and 8 are composite numbers. Such primes fall "out of" any Cunningham chain of first kind (or generate Cunningham chains of 0-length).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)
C. K. Caldwell, Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.

Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* Zak Seidov, Mar 09 2013 *)

is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ Charles R Greathouse IV, Jan 16 2013

a(n) ~ n log n. - Charles R Greathouse IV, Jan 16 2013

A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1178609, 1333889, 1598699, 1806089, 1958249, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4444829, 4664249, 4894889, 4897709, 5132999, 5215499, 5238179, 6026309, 6059519, 6088529, 6490769, 6676259

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special terms of A059453. Not identical to A023330 of which 1122659, 2164229, 2329469, ..., etc. are omitted since they have exact length 7 or larger.

Unsafe primes starting complete chains of length 6.

			89 is a term because (89-1)/2 = 44 and 64*89+63 = 5759 = 13*443 are composites, while 89, 179, 359, 719, 1439, and 2879 are primes.
1122659 is not a term because it initiates a chain of length 7.
4658939 is not a term because (4658939-1)/2 = 2329469 is prime. - _Sean A. Irvine_, Oct 09 2022

Amiram Eldar, Table of n, a(n) for n = 1..1000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Cf. A005602, A053176, A059500, A057331, A059688, A059690.

Entry revised by N. J. A. Sloane Apr 01 2006

a(12) onward corrected and extended by Sean A. Irvine, Oct 09 2022

A110024 Smallest primes starting a complete three iterations Cunningham chain of the second kind.

Original entry on oeis.org

2131, 2311, 6211, 7411, 10321, 18121, 22531, 23011, 24391, 29671, 31771, 35311, 41491, 46411, 54601, 56311, 60331, 61381, 67651, 78031, 85381, 96931, 99871, 109471, 126001, 134731, 156691, 162451, 165331, 170851, 185131, 205171, 224401

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The word "complete" indicates each chain is exactly 4 primes long (i.e., the chain cannot be a subchain of another one). Other sequences give also primes included in longer chains ("starting" them or not).

Terms computed by Gilles Sadowski.

			2311 is here because, through the operator <*2-1> of the chains of the second kind,
2311 -> 4621 -> 9241 -> 18481 and the chain ends here (with this operator).

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Edited and extended by R. J. Mathar, May 08 2009

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

David W. Wilson

nonn

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Additional comments from Labos Elemer, Jul 23 2003

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

Original entry on oeis.org

17, 43, 67, 71, 101, 103, 109, 127, 137, 149, 151, 163, 181, 197, 223, 241, 257, 269, 283, 311, 317, 349, 353, 373, 389, 401, 409, 433, 449, 461, 463, 487, 521, 523, 557, 569, 571, 599, 617, 631, 643, 647, 677, 701, 709, 739, 751, 769, 773, 787, 797, 821

Offset: 1

Alexandre Wajnberg, Sep 01 2005

The condition that neither 2p - 1 nor 2p + 1 be prime is equivalent to ((p-1) mod 3 = 0) or ((p+1) mod 3 = 0). For example, the prime p = 2^607 - 1 is not in this sequence because p + 1 mod 3 = 2. - Washington Bomfim, Oct 30 2009

			a(1) = 17 is here because 17 * 2 + 1 = 35, 17 * 2 - 1 = 33; (17+1)/2 = 9, (17-1)/2 = 8: four composite numbers.

Chris Caldwell's Prime Glossary, Cunningham chains.
Douglas S. Stones, On prime chains, arXiv:0908.2166 [math.NT] [From _Washington Bomfim_, Oct 30 2009, edited by _R. J. Mathar_, Mar 01 2010]

Cf. A005385, A005383, A060254, A005384, A005382, A068497, A059455, A059762, A057326, A023272, A023302, A059764, A057328, A023330, A005602, A064812, A005603.

nonCunninghamPrimes = {}; Do[p = Prime[n]; If[!PrimeQ[2p - 1] && !PrimeQ[2p + 1] && !PrimeQ[(p - 1)/2] && !PrimeQ[(p + 1)/2], AppendTo[nonCunninghamPrimes, p]], {n, 6!}]; nonCunninghamPrimes (* Vladimir Joseph Stephan Orlovsky, Mar 22 2009 *)

Corrected and extended by Ray Chandler, Sep 02 2005

Replaced link to cached arXiv URL with link to the abstract - R. J. Mathar, Mar 01 2010

cp's OEIS Frontend

A059456 Unsafe primes: primes not in A005385.

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A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

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A059763 Primes starting a Cunningham chain of the first kind of length 4.

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A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

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A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

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A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

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A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

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A110024 Smallest primes starting a complete three iterations Cunningham chain of the second kind.