A059761 -id:A059761 - cp's OEIS frontend

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

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

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

A110022 Primes starting a Cunningham chain of the second kind of length 5.

Original entry on oeis.org

1531, 6841, 15391, 44371, 57991, 83431, 105871, 145021, 150151, 199621, 209431, 212851, 231241, 242551, 291271, 319681, 346141, 377491, 381631, 451411, 481021, 506791, 507781, 512821, 537811, 588871, 680431, 727561, 749761, 782911, 787711

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The definition indicates that each chain is exactly 5 primes long (i.e. the chain cannot be a subchain of a longer one). That's why this sequence is different from A057328 which gives also primes included in longer chains (thus not "starting" them), as 16651, starting a seven primes chain, or 33301, second prime of the same seven primes chain.

			6841 is here because: 6841 through <2p-1> -> 13681-> 27361-> 54721-> 109441 and the chain ends here since 2*109441-1=13*113*149 is composite.

Chris Caldwell's Prime Glossary, Cunningham chains.
G. Löh, Long chains of nearly doubled primes, Math. Comp. vol. 53 no. 188 (1989) pp 751-759.

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

isA110022 := 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 4 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 200000 do p := ithprime(i) ; if isA110022(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Jul 23 2008

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

A110056 Least prime that ends a complete Cunningham chain (of the first kind) of length n.

Original entry on oeis.org

13, 7, 167, 4079, 47, 2879, 71850239, 2444789759, 21981381119, 13357981992959, 681004115066879, 1136001594224639, 16756459239477534719, 781558105952602767359

Offset: 1

Alexandre Wajnberg, Sep 04 2005

"Complete" means that this chain is not part of a longer Cunningham chain of the first kind.
Next term is greater than 4*10^17.
A005602(13)-> 8181864863026139 -> ... -> a(13) = 16756459239477534719. [From Washington Bomfim, Oct 21 2009]

			41->83->167 is a Cunningham chain of the first kind. It is complete because neither (41-1)/2 nor 2*167+1 is prime. It is the first such chain of three primes, so a(3) = 167.

Chris Caldwell's Prime Glossary, Cunn ingham chains.

Cf. A005384, A005385, A007700, A023272, A023302, A023330, A059452, A057326, A059455, A059761, A059762, A059763, A059764, A074313.
Cf. A110059 for Cunningham chains of the second kind.
Cf. A005602 [From Washington Bomfim, Oct 21 2009]

Edited and extended by David Wasserman, Aug 08 2006

a(13) and a(14) from Washington Bomfim, Oct 21 2009

A110059 Smallest prime ending a complete Cunningham chain of the second kind (2x-1) of length n.

Original entry on oeis.org

11, 13, 5, 17041, 24481, 12338881, 1065601, 1985902081, 219416417281, 105230562877441, 1422461638625281, 444124661486837761, 3105111850422067201

Offset: 1

Alexandre Wajnberg, Sep 04 2005

nonn

"Complete" means that the chain is not part of a longer chain.

A005603 has the first prime of each chain.

			a(4)=17041 because 2131,4261,8521,17041 are prime, but the preceding and following numbers (1066,34081) are not.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A005382, A005384, A057326, A059761, A005603.

Some terms computed by Gilles Sadowski.

Edited by Don Reble, May 16 2006

A110025 Smallest primes starting a complete three iterations Cunningham chain of the first or second kind.

Original entry on oeis.org

509, 1229, 1409, 2131, 2311, 2699, 3539, 6211, 6449, 7411, 10321, 10589, 11549, 11909, 12119, 17159, 18121, 19709, 19889, 22349, 22531, 23011, 24391, 26189, 27479, 29671, 30389, 31771, 35311, 41491, 43649, 46411, 54601, 55229, 56311

Offset: 1

Alexandre Wajnberg, Sep 03 2005

Terms computed by Gilles Sadowski.

			1409 is here because, through the operator <2p+1> for chains of the first kind, 1409 -> 2819 -> 5639 -> 11279 and the chain ends here.
2131 is here because, through the operator <2p-1> for chains of the second kind, 2131 -> 4261 -> 8521 -> 17041 and the chain ends here.

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.

Union of A059763 and A110024. [R. J. Mathar, May 08 2009]

Edited by R. J. Mathar, May 08 2009

A110027 Smallest primes starting a complete four iterations Cunningham chain of the first or second kind.

Original entry on oeis.org

2, 1531, 6841, 15391, 44371, 53639, 53849, 57991, 61409, 66749, 83431, 105871, 143609, 145021, 150151, 167729, 186149, 199621, 206369, 209431, 212851, 231241, 242551, 268049, 291271, 296099, 319681, 340919, 346141, 377491, 381631, 422069

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The word "complete" indicates each chain is exactly 5 primes long (i.e., the chain cannot be a subchain of another one).

Terms computed by Gilles Sadowski.

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.

Union of A059764 and A110022 . [R. J. Mathar, May 08 2009]

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

A176379 The smallest prime q which stays prime through at least two iterations of q -> := 2 * q + prime(n+1).

Original entry on oeis.org

2, 7, 2, 31, 2, 7, 11, 7, 19, 5, 5, 19, 2, 13, 13, 61, 11, 17, 61, 5, 5, 7, 139, 5, 19, 2, 103, 29, 7, 2, 109, 7, 59, 31, 41, 5, 5, 127, 13, 31, 5, 109, 2, 7, 41, 11, 2, 7, 101, 67, 79, 5, 31, 13, 37, 19, 11, 2, 109, 53, 7, 2, 19, 2, 127, 29, 5, 13, 59, 7, 19, 47, 47, 11, 13, 79, 17, 19, 89, 619

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 16 2010

Such q are generalized Cunningham primes: prime numbers p(1), ..., p(n):
q, f(q) = 2 * q + prime(n+1), f(f(q)) = 4 * q + 3 * prime(n+1) must be primes.
n = 0 is omitted as the first iteration 2 * q + prime(1) = 2 * (q+1) is generally even.
List of (q, first iteration, second iteration):
(2,7,17) (7,19,43) (2,11,29) (31,73,157) (2,17,47)
(7,31,79) (11,41,101) (7,37,97) (19,67,163) (5,41,113)
(5,47,131) (19,79,199) (2,47,137) (13,73,193) (13,79,211)
(61,181,421) (11,83,227) (17,101,269) (61,193,457) (5,83,239)
(5,89,257) (7,97,277) (139,367,823) (5,107,311) (19,139,379)
(2,107,317) (103,313,733) (29,167,443) (7,127,367) (2,131,389)
(109,349,829) (7,151,439) (59,257,653) (31,211,571) (41,233,617)
(5,167,491) (5,173,509) (127,421,1009) (13,199,571) (31,241,661)
(5,191,563) (109,409,1009) (2,197,587) (7,211,619) (41,281,761)
(11,233,677) (2,227,677) (7,241,709) (101,431,1091) (67,367,967)

			n=1, prime(n+1) = 3: checking q=2: 2 * 2 + 3 = 7, 2 * 7 + 3 = 17, q=2 is first term.
n=2: checking q=7: 2 * 7 + 5 = 19, 2 * 19 + 5 = 43, 7 is 2nd term.
n=3: checking q=2: 2 * 2 + 7 = 11, 2 * 11 + 7 = 29, 2 is 3rd term.

Joe Buhler, Algorithmic Number Theory, Third International Symposium, ANTS-III, Springer New York, 1998.
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1994.
Paulo Ribenboim, Die Welt der Primzahlen, Geheimnisse und Rekorde, Springer-Verlag GmbH & Co. KG, 2006.

Cf. A005602, A059761, A059762, A110024.

A176379 := proc(n)
    pk1 := ithprime(n+1) ;
    for pidx from 1 do
        p := ithprime(pidx) ;
        pitr := 2*p+pk1 ;
        if not isprime(pitr) then
            next ;
        end if;
        pitr := 2*pitr+pk1 ;
        if not isprime(pitr) then
            next ;
        else
            return p ;
        end if;
    end do:
end proc:
seq(A176379(n),n=1..20) ; # R. J. Mathar, May 21 2025

a(n) = smallest prime q such that 2*q+prime(n+1) is prime and 2*(2*q+prime(n+1))+prime(n+1) is also prime.

cp's OEIS Frontend

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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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.

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A110022 Primes starting a Cunningham chain of the second kind of length 5.

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A110056 Least prime that ends a complete Cunningham chain (of the first kind) of length n.

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A110059 Smallest prime ending a complete Cunningham chain of the second kind (2x-1) of length n.

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A110025 Smallest primes starting a complete three iterations Cunningham chain of the first or second kind.

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