A023302 -id:A023302 - cp's OEIS frontend

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

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

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

A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

Original entry on oeis.org

5, 2, 5, 2, 5, 0, 0, 0, 5, 2, 0, 0, 3, 0, 5, 2, 2, 0, 0, 0, 0, 0, 3, 6, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 6, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Labos Elemer, Feb 06 2001

nonn

The length of a chain is measured by the total number of terms including the end points. a(n)=0 means that prime(n) is neither Sophie Germain nor a safe prime (i.e. it is in A059500).

			For all of {2,5,11,23,47}, i.e. at positions {j}={1,3,5,9,15} a(j)=5. Similarly for indices of all terms in {89,...,5759} a(i)=6. No chains are intelligible with length = 1 because the minimal chain enclose one Sophie Germain and also one safe prime. Dominant values are 0 and 2.

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

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A059500.

Offset and a(5) corrected by Sean A. Irvine, Oct 01 2022

A059767 Initial (unsafe) primes of Cunningham chains of first type with length exactly 7.

Original entry on oeis.org

1122659, 2164229, 2329469, 10257809, 10309889, 12314699, 14030309, 14145539, 23103659, 24176129, 28843649, 37088729, 42389519, 49160099, 50785439, 62800169, 68718059, 88174049, 95831189, 105388169, 121255889, 138140729, 155439419, 159938459, 173285999

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special primes from A059453.

Primes p such that (2^k)*p+(2^k)-1 is also prime for k = 0, 1, 2, 3, 4, 5, 6 and is composite for k = -1 and k = 7.

			C7 prime chain is generated from prime a(10) = 24176129 with 2p+1 iterations: 24176129, 48352259, 96704519, 193409039, 386818079, 773636159, 1547272319, 3094544639.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 178 (Rev. ed. 1997).

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

Transpose[Select[{#, Length[NestWhileList[2#+1&, #, PrimeQ]]-1}&/@ Prime[Range[PrimePi[24177000]]], #[[2]]>6&]][[1]]
Select[Prime[Range[10^6]], PrimeQ[a1=2*#+1]&&PrimeQ[a2=2*a1+1]&&PrimeQ[a3=2*a2+1]&&PrimeQ[a4=2*a3+1]&&PrimeQ[a5=2*a4+1]&&PrimeQ[a6=2*a5+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=n%30==29 && isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) && isprime(32*n+31) && isprime(64*n+63) && !isprime(n\2) && !isprime(128*n+127) \\ Charles R Greathouse IV, Dec 01 2016

Corrected and extended by Harvey P. Dale, Jul 10 2002
More terms from Vladimir Joseph Stephan Orlovsky, Jan 17 2009
Corrected by John Cerkan, Nov 30 2016

A162019 Double-safe primes which are also double-Sophie Germain primes.

Original entry on oeis.org

11, 359, 719, 214559, 215399, 245639, 253679, 266999, 507359, 508559, 574439, 670919, 744599, 825479, 1017119, 1072199, 1184399, 1363679, 1621079, 1688279, 1786439, 2156039, 2377799, 2429279, 2633399, 2684999, 2900039, 3103799

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 24 2009

nonn

The intersection of the primes in A066179 and those in A007700: they remain prime after each of two successive applications of the substitution p->(p-1)/2, and remain prime after each two successive applications of the substitution p->2p+1.

			a(1)=11 is double safe: (11-1)/2=5; (5-1)/2=2, and double Sophie-Germain: 2*11+1=23; 2*23+1=47.

Cf. A007700, A023302, A066179, A157359.

lst={};Do[p=Prime[n];If[PrimeQ[safe=(p-1)/2],If[PrimeQ[(safe-1)/2],If[PrimeQ[sophie=2*p+1],If[PrimeQ[2*sophie+1],AppendTo[lst,p]]]]],{n,3*9!}];lst

a(n) = 4*A023302(n) + 3 = (A157359(n)-3)/4. - R. J. Mathar, Jun 26 2009

Edited by R. J. Mathar, Jun 26 2009

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

A278932 Numbers n such that n remains prime through 6 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

1122659, 2164229, 2329469, 10257809, 10309889, 12314699, 14030309, 14145539, 19099919, 23103659, 24176129, 28843649, 37088729, 38199839, 42389519, 49160099, 50785439, 52554569, 62800169, 68718059, 85864769, 88174049, 95831189, 105109139, 105388169

Offset: 1

John Cerkan, Dec 01 2016

nonn

n, 2*n+1, 4*n+3, 8*n+7, 16*n+15, 32*n+31, and 64*n+63 are primes.

a(n) == 29 (mod 30).

Subsequence of A007700, A023272, A023302, and A023330.

a005408(n) = 2*n+1
count(n) = my(k=n, i=0); while(ispseudoprime(k), k=a005408(k); i++); i
is(n) = count(n) > 6 \\ Felix Fröhlich, Dec 05 2016

A086137 Number of primes between p and p+8 if p is prime, i.e., number of primes between 8+A023202(n) and A023202(n).

Original entry on oeis.org

2, 2, 2, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 1, 0, 1

Offset: 1

Labos Elemer, Jul 29 2003

nonn

			a(n)=0,1,2 correspond to {p,p+8} prime-pairs either consecutive or pairs with various d-patterns as follows:
a(n)=0 to 89[8]97; a(n)=1 for 29[2,6]37, 53[6,2];
a(n)=2 for 101[2,4,2]109 and once to 3[2,2,4]11.

Cf. A023302, A046133, A048136, A052165.

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 3, 2, 509, 2, 89, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one). a(1)-a(8) computed by Gilles Sadowski.

			a(1)=11 because 2, 3, 5 and 7 are included in longer chains than one prime long; and 11 (although included in a <2p+1> chain) has no prime connection through <2p-1>.
a(2)=3 because 3 begins (through 2p+1>) the first complete two primes chain: 3-> 7 (even if 3 and 7 are also part of two others chains, but through <2p-1>).
a(3)=2 because (although 2 begins also a five primes chain through <2p+1>) it begins, through <2p-1>, the first complete three primes chain encountered: 2->3->5.

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, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

a(n) = min(A005602(n), A005603(n)). - R. J. Mathar, Jul 23 2008

a(8)-a(13) via A005602, A005603 from R. J. Mathar, Jul 23 2008

a(14)-a(15) via A005602, A005603 from Jason Yuen, Sep 03 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A059767 Initial (unsafe) primes of Cunningham chains of first type with length exactly 7.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A162019 Double-safe primes which are also double-Sophie Germain primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A278932 Numbers n such that n remains prime through 6 iterations of function f(x) = 2x + 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.