A059453 -id:A059453 - cp's OEIS frontend

A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

41, 1031, 1451, 1481, 1511, 1811, 1889, 1901, 1931, 3449, 3491, 3821, 3911, 5081, 5441, 5849, 6101, 6131, 7151, 7349, 7901, 8969, 9221, 10691, 10709, 11171, 11471, 11801, 12101, 12821, 12959, 13229, 14009, 14249, 14321, 14669, 14741, 15161

Offset: 1

Labos Elemer, Feb 20 2001

nonn

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

			41 is a term because 20 and 325 are composites, and 41, 83, and 167 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains.
Eric Weisstein's World of Mathematics, Cunningham Chain.

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

ipccQ[n_]:=Module[{c=(n-1)/2},PrimeQ[NestList[2#+1&,c,4]]=={False, True, True, True, False}]; Select[Prime[Range[2000]],ipccQ] (* Harvey P. Dale, Nov 10 2014 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Offset corrected by Amiram Eldar, Jul 15 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 *)

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

A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 7, 13, 20, 31, 52, 83, 142, 242, 412, 742, 1308, 2294, 4040, 7327, 13253, 24255, 44306, 81700, 150401, 277335, 513705, 954847, 1780466, 3325109, 6224282, 11676337, 21947583, 41327438

Offset: 1

Labos Elemer, Feb 06 2001

			a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.

c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]

from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
    s = 1
    for n in count(2):
        yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022

Edited and extended by Robert G. Wilson v, Nov 23 2002
Title and a(30)-a(31) corrected, and a(32) from Sean A. Irvine, Oct 02 2022
a(33)-a(34) from Michael S. Branicky, Oct 09 2022

A053176 Primes p such that 2p+1 is composite.

Original entry on oeis.org

7, 13, 17, 19, 31, 37, 43, 47, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 181, 193, 197, 199, 211, 223, 227, 229, 241, 257, 263, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 383

Offset: 1

Enoch Haga, Feb 29 2000

Primes not in A005384 = non-Sophie Germain primes.
Also, numbers n such that odd part of A005277(n) is prime. Proof by John Renze, Sep 30 2004
Sequence gives primes p such that B(2p) has denominator 6, where B(2n) are the Bernoulli numbers. - Benoit Cloitre, Feb 06 2002
Sequence gives all n such that the equation phi(x)=2n has no solution. - Benoit Cloitre, Apr 07 2002
A010051(a(n))*(1-A156660(a(n))) = 1; subsequence of A138887. - Reinhard Zumkeller, Feb 18 2009
Mersenne prime exponents > 3 must be in the union of this sequence and (A002144). - Roderick MacPhee, Jan 12 2017

			17 is a term because 2*17 + 1 = 35 is composite.

T. D. Noe, Table of n, a(n) for n = 1..10000
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.

Cf. A005384, A005385, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156543, A156542.

[p: p in PrimesUpTo(12200) | not IsPrime(2*p+1)]; // Vincenzo Librandi, Jun 18 2015

Select[Prime[Range[1000]], ! PrimeQ[2 # + 1] &] (* Vincenzo Librandi, Jun 18 2015 *)

list(lim)=select(p->!isprime(2*p+1),primes(primepi(lim))) \\ Charles R Greathouse IV, Jul 25 2011

a(n) ~ n log n. - Charles R Greathouse IV, Feb 20 2012

A059455 Safe primes which are also Sophie Germain primes.

Original entry on oeis.org

5, 11, 23, 83, 179, 359, 719, 1019, 1439, 2039, 2063, 2459, 2819, 2903, 2963, 3023, 3623, 3779, 3803, 3863, 4919, 5399, 5639, 6899, 6983, 7079, 7643, 7823, 10163, 10799, 10883, 11699, 12203, 12263, 12899, 14159, 14303, 14699, 15803, 17939

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Primes p such that both (p-1)/2 and 2*p+1 are prime.
Except for 5, all are congruent to 11 modulo 12.
Primes "inside" Cunningham chains of first kind.
Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 18 2012
See A162019 for the subset of a(n) that are "reproduced" by the application of the transformations (a(n)-1)/2 and 2*a(n)+1 to the set a(n). - Richard R. Forberg, Mar 05 2015

			83 is a term because it is prime and 2*83+1 = 167 and (83-1)/2 = 41 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A005385.

Cf. A053176, A059452, A059453, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.

[p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Oct 31 2014

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[2*p+1], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 02 2008 *)
Select[Prime[Range[1000]], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* requires Mathematica 10+; Feras Awad, Dec 19 2018 *)

forprime(p=2,1e5,if(isprime(p\2)&&isprime(2*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from itertools import count, islice
from sympy import isprime, prime
def A059455_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059455_list = list(islice(A059455_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*A156659(a(n)) = 1; A156877 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

A059452 Safe primes (A005385) that are not Sophie Germain primes.

Original entry on oeis.org

7, 47, 59, 107, 167, 227, 263, 347, 383, 467, 479, 503, 563, 587, 839, 863, 887, 983, 1187, 1283, 1307, 1319, 1367, 1487, 1523, 1619, 1823, 1907, 2027, 2099, 2207, 2447, 2579, 2879, 2999, 3119, 3167, 3203, 3467, 3947, 4007, 4079, 4127, 4139, 4259, 4283

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 7, these primes are congruent to 11 modulo 12.

Terminal primes in complete Cunningham chains of first kind, i.e., the chains cannot be continued from these primes.

			347 is a term because 173 is a prime but 695 is not.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chain.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]

Intersection of A005385 and A053176.

Cf. A005384, A059452, A059453, A059454, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660.

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 *)

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

from itertools import count, islice
from sympy import isprime, prime
def A059452_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and not isprime(p<<1|1),(prime(i) for i in count(1)))
A059452_list = list(islice(A059452_gen(),10)) # Chai Wah Wu, Jul 12 2022

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

Broken link updated by R. J. Mathar, Apr 12 2010

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

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

A075712 Rearrangement of primes into Germain groups (or Cunningham chains).

Original entry on oeis.org

2, 5, 11, 23, 47, 3, 7, 13, 17, 19, 29, 59, 31, 37, 41, 83, 167, 43, 53, 107, 61, 67, 71, 73, 79, 89, 179, 359, 719, 1439, 2879, 97, 101, 103, 109, 113, 227, 127, 131, 263, 137, 139, 149, 151, 157, 163, 173, 347, 181, 191, 383, 193, 197, 199, 211, 223, 229, 233

Offset: 1

Zak Seidov, Oct 03 2002

In each group, p(i+1) = 2*p(i)+1.

The groups are also known as Cunningham chains of the first kind.

			The groups are:
{2, 5, 11, 23, 47},
{3, 7},
{13},
{17},
{19},
{29, 59},
{31},
{37},
{41, 83, 167},
{43},
{53, 107},
{61},
{67},
{71},
{73},
{79},
{89, 179, 359, 719, 1439, 2879},
{97},
{101},
{103},
{109},
{113, 227},
{127},
{131, 263},
{137},
{139},
...

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

Cf. A005384, A059452, A059453, A059455, A059456, A053176.
See also A181697.
See A059456 for initial terms, A338945 for lengths.

Block[{a = {2}, j = 1, k, p}, Do[k = j; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] + 1], While[! FreeQ[a, Set[p, Prime[k]]], k++]; j++; Set[a, Append[a[[1 ;; -2]], p]]], 10^3]; a] (* Michael De Vlieger, Nov 17 2020 *)

first(n) = my(res=List([2,5,11,23,47])); forprime(p=3, oo, if(!isprime((p-1)>>1), listput(res,p); c = 2*p+1; while(isprime(c), listput(res,c); c=2*c+1)); if(#res>n,return(res))); res \\ David A. Corneth, Nov 13 2021

Edited by N. J. A. Sloane, Nov 13 2021

More terms from David A. Corneth, Nov 13 2021

cp's OEIS Frontend

A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

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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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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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A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

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A053176 Primes p such that 2p+1 is composite.

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A059455 Safe primes which are also Sophie Germain primes.

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A059452 Safe primes (A005385) that are not Sophie Germain primes.

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