A057327 -id:A057327 - cp's OEIS frontend

A005382 Primes p such that 2p-1 is also prime.

2, 3, 7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011

Offset: 1

N. J. A. Sloane

Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - Benoit Cloitre, Jan 25 2002
Let q = 2n-1. For these n (and q), the sum of two cyclotomic polynomials can be written as a product of cyclotomic polynomials and as a cyclotomic polynomial in x^2: Phi(q,x) + Phi(2q,x) = 2 Phi(n,x) Phi(2n,x) = 2 Phi(n,x^2). - T. D. Noe, Nov 04 2003
Primes in A006254. - Zak Seidov, Mar 26 2013
If a(n) is in A168421 then A005383(n) is a twin prime with a Ramanujan prime, A005383(n) - 2. If this sequence has an infinite number of terms in A168421, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
Records subsequence of A023509 (n >= 2). - David James Sycamore, May 05 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Ajeet Kumar, Subhamoy Maitra, and Chandra Sekhar Mukherjee, On approximate real mutually unbiased bases in square dimension, Cryptography and Communications (2020) Vol. 13, 321-329.
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025. See p. 8.
Marius Tărnăuceanu, Arithmetic progressions in finite groups, arXiv:2003.10060 [math.GR], 2020.
Wikipedia, Cunningham chain

Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.
Cf. A010051, A000040, A053685 (subsequence), A006254.
Cf. A023509.

a005382 n = a005382_list !! (n-1)
a005382_list = filter
   ((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
-- Reinhard Zumkeller, Oct 03 2012

[n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

f := proc(Q) local t1,i,j; t1 := []; for i from 1 to 500 do j := ithprime(i); if isprime(2*j-Q) then t1 := [op(t1),j]; fi; od: t1; end; f(1);
# second Maple program:
q:= p-> andmap(isprime, [p, 2*p-1]):
select(q, [$2..2500])[];  # Alois P. Heinz, Dec 16 2024

Select[Prime[Range[300]], PrimeQ[2#-1]&]

select(p->isprime(2*p-1),primes(500)) \\ Charles R Greathouse IV, Apr 26 2012

forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

a(n) = A129521(n) / A005383(n). - Reinhard Zumkeller, Apr 19 2007

a(n) = (A005383(n) + 1)/2. - Zak Seidov, Nov 04 2010

A005383 Primes p such that (p+1)/2 is prime.

Original entry on oeis.org

3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset: 1

N. J. A. Sloane

Also, n such that sigma(n)/2 is prime. - Joseph L. Pe, Dec 10 2001; confirmed by Vladeta Jovovic, Dec 12 2002
Primes that are followed by twice a prime, i.e., are followed by a semiprime. (For primes followed by two semiprimes, see A036570.) - Zak Seidov, Aug 03 2013, Dec 31 2015
If A005382(n) is in A168421 then a(n) is a twin prime with a Ramanujan prime, A104272(k) = a(n) - 2. - John W. Nicholson, Jan 07 2016
Starting with 13 all terms are congruent to 1 mod 12. - Zak Seidov, Feb 16 2017
Numbers n such that both n and n+12 are terms are 61, 661, 1201, 4261, 5101, 6121, 6361 (all congruent to 1 mod 60). - Zak Seidov, Mar 16 2017
Primes p for which there exists a prime q < p such that 2q == 1 (mod p). Proof: q = (p + 1)/2. - David James Sycamore, Nov 10 2018
Prime numbers n such that phi(sigma(2n)) = phi(2n), excluding n=3 and n=5; as well as phi(sigma(3n)) = phi(3n), excluding n=3 only. - Richard R. Forberg, Dec 22 2020

			Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - _Zak Seidov_, Nov 19 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Benoit Cloitre, On the fractal behavior of primes, 2011.

Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.
A subsequence of A000040 which has A036570 as subsequence.
Cf. A005385, A010051, A065091, A048161, A036570.

a005383 n = a005383_list !! (n-1)
a005383_list = [p | p <- a065091_list, a010051 ((p + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Nov 06 2012

LIMIT = 8000 % Find all members of A005383 less than LIMIT A = primes(LIMIT); n = length(A); %n is number of primes less than LIMIT B = 2*A - 1; C = ones(n, 1)*A; %C is an n X n matrix, with C(i, j) = j-th prime D = B'*ones(1, n); %D is an n X n matrix, with D(i, j) = (i-th prime)*2 - 1 [i, j] = find(C == D); A(j)

[n: n in [1..3300] | IsPrime(n) and IsPrime((n+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

for n to 300 do
  X := ithprime(n);
Y := ithprime(n+1);
Z := 1/2 mod Y;
  if isprime(Z) then print(Y);
end if:
end do:
# David James Sycamore, Nov 11 2018

Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)

A005383_list(n) = select(m->isprime(m\2+1),primes(n)[2..n]) \\ Charles R Greathouse IV, Sep 25 2012

from sympy import isprime
[n for n in range(3, 5000) if isprime(n) and isprime((n + 1)//2)]
# Indranil Ghosh, Mar 17 2017

[n for n in prime_range(3, 1000) if is_prime((n + 1) // 2)]
# F. Chapoton, Dec 17 2019

a(n) = A129521(n)/A005382(n). - Reinhard Zumkeller, Apr 19 2007
A000035(a(n))*A010051(a(n))*A010051((a(n)+1)/2) = 1. - Reinhard Zumkeller, Nov 06 2012
a(n) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012
a(n) = A005382(n) + phi(A005382(n)) = A005382(n) + A000010(A005382(n)). - Torlach Rush, Mar 10 2014

More terms from David Wasserman, Jan 18 2002

Name changed by Jianing Song, Nov 27 2021

A057326 First member of a prime triple in a 2p-1 progression.

Original entry on oeis.org

2, 19, 79, 331, 439, 499, 619, 829, 1069, 1279, 1531, 2089, 2131, 2179, 2311, 2791, 3019, 3061, 3109, 3181, 3769, 4159, 4231, 4261, 4621, 4639, 4861, 4951, 5419, 5749, 6121, 6211, 6709, 6841, 7369, 7411, 7561, 7639, 8209, 8629, 9109, 9199, 9319, 9739, 10321, 10831

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

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

			Triplets are (2,3,5), (19,37,73), (79,157,313), (331,661,1321), ...

Vincenzo Librandi, Table of n, a(n) for n = 1..2100
Index entries for sequences related to primes in arithmetic progressions

Cf. (A005382 and A005383), A057327, A057328, A057329, A057330, A005603.

[p: p in PrimesUpTo (10000) | IsPrime(2*p-1) and IsPrime(4*p-3)]; // K. D. Bajpai, Jun 26 2017

select(p -> andmap(isprime,[p, 2*p-1, 4*p-3]), [seq(p, p=0..10000)]); # K. D. Bajpai, Jun 26 2017

Select[Prime[Range[1500]],And@@PrimeQ[NestList[2#-1&,#,2]]&] (* Harvey P. Dale, Dec 09 2011 *)

forprime(p= 1, 100000, if(isprime(2*p-1) && isprime(4*p-3), print1(p, ", "))); \\ K. D. Bajpai, Jun 26 2017

Offset set to 1 by Michel Marcus, Jul 02 2017

A057330 First member of a prime n-tuplet in a 2p-1 progression.

Original entry on oeis.org

2, 2, 2, 1531, 1531, 16651, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 69257563144280941, 69257563144280941, 3203000719597029781

Offset: 1

Patrick De Geest, Aug 15 2000

Initial terms of A000040, A005382, A057326, A057327, A057328, A057329.

Dirk Augustin, Smallest existing Cunningham Chains of given length
Chris Caldwell, The Prime Glossary: Cunningham chain
Tony Forbes, Cunningham Chain of length 16, Dec 05 1997
Index entries for sequences related to primes in arithmetic progressions

A005603 Smallest prime beginning a complete Cunningham chain (of the second kind) of length n.

Original entry on oeis.org

11, 7, 2, 2131, 1531, 385591, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 107588900851484911, 69257563144280941, 3203000719597029781

Offset: 1

N. J. A. Sloane

The chain begins with a prime number p; next term p' (a prime) is produced forming 2p-1; next term p"=2p'-1, etc. "Complete" means that each chain is exactly n primes long (i.e. the chain cannot be a subchain of another one). That is why this sequence is slightly different from A064812, where the 6th term (33301) is smaller than here (385591) but is the second one of a seven primes sequence and therefore doesn't *start* a sequence.

According to Augustin's web site, the numbers 107588900851484911, 69257563144280941, 3203000719597029781 are also in the sequence. - Dmitry Kamenetsky, May 14 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Dirk Augustin, Cunningham Chain records.
C. K. Caldwell, Cunningham chain.
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.

See A064812 for another version.

Cf. (A005382 and A005383), A057326, A057327, A057328, A057329, A057330, A057331, A005602.

6th term corrected from 385591 on Feb 23 1995, at Robert G. Wilson v's suggestion
a(14) and a(15) found by Paul Jobling (Paul.Jobling(AT)WhiteCross.com) [Oct 23 2000]
a(6) reverted to original value by Sean A. Irvine, Jul 10 2016
a(16) from Augustin's page, comment corrected by Jens Kruse Andersen, Jun 14 2014
Edited by N. J. A. Sloane, Nov 03 2018 at the suggestion of Georg Fischer, Nov 03 2018, merging a duplicate entry with this one.
In Augustin's web page there are 7 or so more terms which could be added here, or alternatively used to create a b-file. - Georg Fischer, Nov 03 2018

A057328 First member of a prime 5-tuple in a 2p-1 progression.

Original entry on oeis.org

1531, 6841, 15391, 16651, 33301, 44371, 57991, 66601, 83431, 105871, 145021, 150151, 165901, 199621, 209431, 212851, 231241, 242551, 291271, 319681, 331801, 346141, 377491, 381631, 385591, 445741, 451411, 478801, 481021, 506791, 507781

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

Numbers n such that n remains prime through 4 iterations of function f(x) = 2x - 1.

			Quintuplets are (1531, 3061, 6121, 12241, 24481), (6841, 13681, 27361, 54721, 109441), ...

Cf. A005382, A005383, A057326, A057327, A057329, A057330, A005603.

[ p: p in PrimesUpTo(6*10^5) | forall{q: k in [1..4] | IsPrime(q) where q is 2^k*(p-1)+1} ];  // Bruno Berselli, Nov 23 2011

pQ[n_] := And @@ PrimeQ[NestList[2 # - 1 &, n, 4]]; t = {}; Do[p = Prime[n]; If[pQ[p], AppendTo[t, p]], {n, 42500}]; t (* Jayanta Basu, Jun 17 2013 *)
Select[Prime[Range[50000]],AllTrue[Rest[NestList[2#-1&,#,4]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2019 *)

A057329 First member of a prime sextuplet in a 2p-1 progression.

Original entry on oeis.org

16651, 33301, 165901, 331801, 385591, 445741, 478801, 580471, 1203121, 1768441, 1943371, 2041201, 2131141, 2240941, 2340661, 2393431, 2526721, 3277471, 3536881, 3623881, 3880381, 3897631, 4123621, 4415371, 4481881, 5278591

Offset: 1

Patrick De Geest, Aug 15 2000

nonn

Numbers n such that n remains prime through 5 iterations of function f(x) = 2x - 1.

			First sextuplet is (16651,33301,66601,133201,266401,532801).

Index entries for sequences related to primes in arithmetic progressions

Cf. A005382 and A005383; A057326, A057327, A057328, A057330, A005603.

[ p: p in PrimesUpTo(6*10^6) | forall{q: k in [1..5] | IsPrime(q) where q is 2^k*(p-1)+1} ];  // Bruno Berselli, Nov 23 2011

Offset changed by Andrew Howroyd, Aug 13 2024

A181715 Length of the complete Cunningham chain of the second kind starting with prime(n).

Original entry on oeis.org

3, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 17 2010

nonn

Number of iterations x -> 2x-1 needed to get a composite number, when starting with prime(n).
Dickson's conjecture implies that, for every positive integer r, there exist infinitely many n such that a(n) = r. - Lorenzo Sauras Altuzarra, Feb 12 2021
a(n) is the least k such that 2^k * (prime(n)-1) + 1 is composite. Note that a(n) is well defined since 2^(p-1) * (p-1) + 1 is divisible by p for odd primes p. - Jianing Song, Nov 24 2021

			2 -> 3 -> 5 -> 9 = 3^2, so a(1) = 3 and a(2) = 2. - _Jonathan Sondow_, Oct 30 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Michael Penn, Romanian Mathematical Olympiad Problem, Youtube video, 2020.
Wikipedia, Cunningham chain

Cf. A000040, A005382, A005408, A005602, A005603, A181697, A263879, A285700, A285706, A057326, A057327, A057328, A057329, A057330, A064812.

Cf. A137288 (positions of terms > 1).

a := proc(n)
   local c, l:
   c, l := 0, ithprime(n):
   while isprime(l) do c, l := c+1, 2*l-1: od:
   c:
end: # Lorenzo Sauras Altuzarra, Feb 12 2021

Table[p = Prime[n]; cnt = 1; While[p = 2*p - 1; PrimeQ[p], cnt++]; cnt, {n, 100}] (* T. D. Noe, Jul 12 2012 *)
Table[-1 + Length@ NestWhileList[2 # - 1 &, Prime@ n, PrimeQ@ # &], {n, 98}] (* Michael De Vlieger, Apr 26 2017 *)

a(n)= n=prime(n); for(c=1,1e9, is/*pseudo*/prime(n=2*n-1) || return(c))

a(n) < prime(n) for n > 1; see Löh (1989), p. 751. - Jonathan Sondow, Oct 28 2015
max(a(n), A181697(n)) = A263879(n) for n > 2. - Jonathan Sondow, Oct 30 2015
a(n) = A285700(A000040(n)). - Antti Karttunen, Apr 26 2017

Escape clause added to definition by N. J. A. Sloane, Feb 19 2021

Escape clause deleted from definition by Jianing Song, Nov 24 2021

A064812 Smallest prime p such that the infinite sequence {p, p'=2p-1, p''=2p'-1, ...} begins with a string of exactly n primes.

Original entry on oeis.org

5, 3, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 107588900851484911, 69257563144280941

Offset: 1

David Terr, Oct 21 2002

nonn

Chains of length n of nearly doubled primes.

Smallest prime beginning a complete Cunningham chain of length n of the second kind. (For the first kind see A005602.) - Jonathan Sondow, Oct 30 2015

			a(3) = 2 because 2 is the smallest prime such that the sequence {2, 3, 5, 9, ...} begins with exactly 3 primes, where each term in the sequence is twice the preceding term minus 1.

G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Wikipedia, Cunningham chain

A better version of A005603.
Cf. (A005382 and A005383), A057326, A057327, A057328, A057329, A057330, A005602.
Cf. A002808, A006450, A049076-A049081, A050436, A076237-A076239, A050436-A050439, A065860, A181697, A263879.

A336060 Numbers p such that p, 2p-1, 3p-2, 4p-3, 5p-4, 6p-5, 7p-6, 8p-7 are primes.

Original entry on oeis.org

512821, 22240681, 26486461, 99597961, 593009341, 1021157551, 1041298441, 1281196561, 1349985001, 1426148011, 1660907431, 1894757971, 2544774541, 3590232241, 3958824871, 4012463281, 4219580191, 4363137241, 6482599411, 7178651971, 8051820421, 8417194381

Offset: 1

Jeppe Stig Nielsen, Jul 07 2020

nonn

Each term is 1 modulo 210.

The subset p, 2p-1, 4p-3, 8p-7 is a Cunningham chain of the 2nd kind, cf. A057327.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50
Wikipedia, Cunningham chain

Cf. A005382, A057327, A101769, A336059.

Select[Prime[Range[387*10^6]],AllTrue[Table[n #-(n-1),{n,2,8}],PrimeQ]&] (* Harvey P. Dale, Mar 01 2023 *)

cp's OEIS Frontend

A005382 Primes p such that 2p-1 is also prime.

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A005383 Primes p such that (p+1)/2 is prime.

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A057326 First member of a prime triple in a 2p-1 progression.

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A057330 First member of a prime n-tuplet in a 2p-1 progression.

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A005603 Smallest prime beginning a complete Cunningham chain (of the second kind) of length n.

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A057328 First member of a prime 5-tuple in a 2p-1 progression.

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A057329 First member of a prime sextuplet in a 2p-1 progression.