A007700 -id:A007700 - cp's OEIS frontend

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

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

A107021 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7 are all primes.

Original entry on oeis.org

2, 6449, 12119, 19709, 30389, 74699, 107699, 133499, 143609, 167759, 175349, 206369, 210209, 229739, 244589, 254279, 334289, 422069, 528509, 541529, 607319, 641969, 658349, 751529, 810539, 810809, 812849, 926669, 934259, 956909, 968729

Offset: 1

Zak Seidov, May 09 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A107024: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 all prime; A107023: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 all prime; A107022: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 all prime; A107020: p, 2p+1, 4p+3, 6p+5 all prime; A007700: p, 2p+1, 4p+3 all prime; A005384: p, 2p+1 prime (p = Sophie Germain primes).

Cf. A005384, A007700, A107020, A107022, A107023, A107024.

[p: p in PrimesUpTo(1000000)| IsPrime(2*p+1) and IsPrime(4*p+3) and IsPrime(6*p+5) and IsPrime(8*p+7)]; // Vincenzo Librandi, Nov 13 2010

fQ[n_]:=And@@PrimeQ[{2n+1,4n+3,6n+5,8n+7}];Select[Prime@Range@77000,fQ] (* Harvey P. Dale, Dec 16 2010 *)

More terms from Vincenzo Librandi, Apr 01 2010

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

A107020 Primes p such that 2p+1, 4p+3, 6p+5 are all primes.

Original entry on oeis.org

2, 11, 41, 1901, 2459, 5081, 5849, 6131, 6449, 8969, 9221, 10691, 12119, 13229, 14009, 14321, 14669, 15161, 18461, 19709, 20411, 21179, 22271, 23099, 24551, 25601, 30389, 37991, 39419, 41381, 43691, 44699, 52289, 55631, 56081, 58979

Offset: 1

Zak Seidov, May 09 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A107024: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 all prime; A107023: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 all prime; A107022: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 all prime; A107020: p, 2p+1, 4p+3, 6p+5, 8p+7 all prime; A007700: p, 2p+1, 4p+3 all prime; A005384: p, 2p+1 prime (p = Sophie Germain primes).

Cf. A005384, A007700, A107021, A107022, A107023, A107024.

[p: p in PrimesUpTo(1000000)| IsPrime(2*p+1) and IsPrime(4*p+3) and IsPrime(6*p+5) ]; // Vincenzo Librandi, Nov 13 2010

Select[Range[60000],AllTrue[{#,2#+1,4#+3,6#+5},PrimeQ]&] (* James C. McMahon, Feb 09 2024 *)

More terms from Vincenzo Librandi, Apr 01 2010

A107022 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 are all primes.

Original entry on oeis.org

2, 6449, 210209, 244589, 528509, 810539, 968729, 985109, 1316699, 1551899, 1743419, 2832629, 4094999, 4328459, 5608409, 6036869, 7077419, 7939829, 8176979, 8673569, 8789279, 9080189, 9797279, 10122419, 10309889, 10487969

Offset: 1

Zak Seidov, May 09 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A107024: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 all prime; A107023: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 all prime; A107021: p, 2p+1, 4p+3, 6p+5, 8p+7 all prime;A107020: p, 2p+1, 4p+3, 6p+5 all prime; A007700: p, 2p+1, 4p+3 all prime; A005384: p, 2p+1 prime (p = Sophie Germain primes).

Cf. A005384, A007700, A107020, A107021, A107023, A107024.

[p: p in PrimesUpTo(100000000)| IsPrime(2*p+1) and IsPrime(4*p+3) and IsPrime(6*p+5) and IsPrime(8*p+7)and IsPrime(10*p+9)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[700000]],And@@PrimeQ[{2#+1,4#+3,6#+5,8#+7,10#+9}]&] (* Harvey P. Dale, Jun 19 2013 *)
Select[Prime[Range[700000]],AllTrue[Table[2n #+2n-1,{n,5}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 22 2018 *)

More terms from Vincenzo Librandi, Apr 01 2010

A107023 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 are all primes.

Original entry on oeis.org

4094999, 9080189, 10957169, 11148899, 15917579, 19422059, 37267229, 37622339, 58680929, 63196349, 64595369, 66383519, 108463739, 177109379, 186977699, 189997079, 196068179, 228875849, 251891639, 261703889, 271031669, 310143959

Offset: 1

Zak Seidov, May 09 2005, Mar 08 2007

nonn

			a(1) = p = 4094999 is a term because numbers i*p+(i-1), i=2(2)12 8189999,16379999,24569999,32759999,40949999,49139999 are all primes.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A107024: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 all prime; A107022: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 all prime; A107021: p, 2p+1, 4p+3, 6p+5, 8p+7 all prime;A107020: p, 2p+1, 4p+3, 6p+5 all prime; A007700: p, 2p+1, 4p+3 all prime; A005384: p, 2p+1 prime (p = Sophie Germain primes).

Cf. A005384, A007700, A107020, A107021, A107022, A107024.

s={};Do[p=Prime[i]; If[Union[PrimeQ[Table[i*p+(i-1),{i,2,12,2}]]]=={True},AppendTo[s,p]],{i,289435,1236230}];s
With[{t=Table[2n #+(2n-1),{n,6}]},Select[Prime[ Range[ 168*10^5]], AllTrue[ t,PrimeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 14 2018 *)

A107024 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 are all primes.

Original entry on oeis.org

4094999, 9080189, 63196349, 66383519, 177109379, 196068179, 310143959, 389825729, 528083219, 909696059, 937924259, 1080610439, 1318820159, 1342772969, 1824166469, 1921977329

Offset: 1

Zak Seidov, May 09 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A107023: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 all prime; A107022: p, 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 all prime; A107021: p, 2p+1, 4p+3, 6p+5, 8p+7 all prime; A107020: p, 2p+1, 4p+3, 6p+5 all prime; A007700: p, 2p+1, 4p+3 all prime; A005384: p, 2p+1 prime (p = Sophie Germain primes).

Cf. A005384, A007700, A107020-A107023.

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

A231967 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p + 1 and r = 2q + 1.

Original entry on oeis.org

110, 1265, 11891, 568301, 5719229, 46203659, 371436119, 1057570169, 2978731439, 8475105539, 8777935031, 14865764009, 22397944469, 24460553171, 26008879181, 27621202391, 47549400491, 53960155829, 54994829321, 57639193331, 119010782819, 157361958899

Offset: 1

Jaroslav Krizek, Nov 16 2013

nonn

Squarefree numbers of the form p*q*r, where p < q < r = primes with q = 2*p + 1 and r = 2*q + 1; that is, r = 4*p + 3.

			5719229 = 89*179*359, 179 = 2*89 + 1, 359 = 2*179 + 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005117, A000040, A231968, A231969, A231966. Cf. A007700 (first member of a prime triple in a 2p+1 progression).

sfQ[n_]:=Module[{q=2n+1,r},r=2q+1;AllTrue[{q,r},PrimeQ]&& SquareFreeQ[ n*q*r]]; 3#+10#^2+8#^3&/@Select[Prime[Range[400]],sfQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2016 *)

cp's OEIS Frontend

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

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A107021 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7 are all primes.

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Magma

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

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A107020 Primes p such that 2p+1, 4p+3, 6p+5 are all primes.

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A107022 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9 are all primes.

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Magma

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A107023 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11 are all primes.

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Mathematica

A107024 Primes p such that 2p+1, 4p+3, 6p+5, 8p+7, 10p+9, 12p+11, 14p+13 are all primes.

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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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A231967 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p + 1 and r = 2q + 1.