A005602 -id:A005602 - cp's OEIS frontend

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

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

A059456 Unsafe primes: primes not in A005385.

Original entry on oeis.org

2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Labos Elemer, Feb 02 2001

nonn

A010051(a(n))*(1-A156659(a(n))) = 1; subsequence of A156657. - Reinhard Zumkeller, Feb 18 2009
Also, primes p such that p-1 is a non-semiprime. - Juri-Stepan Gerasimov, Apr 28 2010
Conjecture: From the sequence of prime numbers, let 2 and remove the first data iteration of 2*p+1; leave 3 and remove the prime data by the iteration 2*p+1 and we get the sequence. Example for p=2, remove(5,11,23,47); p=3, remove(7); p=13, p=17, p=19, p=23, remove(47); and so on. - Vincenzo Librandi, Aug 07 2010

			31 is here because (31-1)/2=15 is not prime. 2 and 3 are here because 1/2 and 1 are not prime numbers.

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

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

Initial terms for groups in A075712.

Complement[Prime@ Range@ PrimePi@ Max@ #, #] &@ Select[Prime@ Range@ 90, PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, May 01 2016 *)
Select[Prime[Range[100]],PrimeOmega[#-1]!=2&] (* Harvey P. Dale, May 13 2018 *)

is(n)=isprime(n) && !isprime(n\2) \\ Charles R Greathouse IV, May 02 2016

a(n) ~ n log n. - Charles R Greathouse IV, Dec 29 2024

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

Original entry on oeis.org

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

			89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

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 *)
Select[Prime[Range[300]],PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

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

from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(),10)) # Chai Wah Wu, Jul 12 2022

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

A181697 Length of the complete Cunningham chain of the first kind starting with prime(n).

Original entry on oeis.org

5, 2, 4, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 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).
prime(n) is in A005384, i.e., a Sophie Germain prime, iff a(n)>1.
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 -> 5 -> 11 -> 23 -> 47 -> 95 = 5*19, so a(1) = 5, a(3) = 4, a(5) = 3, a(9) = 2, and a(15) = 1. - _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.
Wikipedia, Cunningham chain

Cf. A005602, A181715, A263879.

A063377 Sophie Germain degree of n: number of iterations of n under f(k) = 2k+1 before we reach a number that is not a prime.

Original entry on oeis.org

0, 5, 2, 0, 4, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Reiner Martin, Jul 14 2001

nonn

a(n) >= 1 means that n is prime; a(n) >= 2 means that n is a Sophie Germain prime. Is the Sophie Germain degree always finite? Is it unbounded?
A339579 is an essentially identical sequence from 1981. - N. J. A. Sloane, Dec 24 2020
From Michael S. Branicky, Dec 24 2020: (Start)
All n > 5 with a(n) >= 4 satisfy n == 9 (mod 10).
Proof.  Let f^k(n) denote iterates of 2*k + 1, with f^0(n) = n.
n != 0, 2, 4, 5, 6, or 8 (mod 10), otherwise f^0(n) is not prime, and a(n) = 0.
n != 7 (mod 10) otherwise f^1(n) = 2*n + 1 == 5 (mod 10), not prime, and a(n) <= 1.
n != 3 (mod 10) otherwise f^2(n) = 4*r + 3 == 5 (mod 10), not prime, and a(n) <= 2.
n != 1 (mod 10) otherwise f^3(n) = 8*r + 7 == 5 (mod 10), not prime, and a(n) <= 3.
(End)
From Peter Schorn, Jan 18 2021: (Start)
The Sophie Germain degree is always finite.
Proof. Let f^k(n) denote iterates of 2*k + 1 with closed form f^k(n) = 2^k * n + 2^k - 1.
There are three cases for n:
1. If n is not a prime then f^0(n) = n is composite.
2. If n = 2 then f^5(2) = 95 is composite.
3. If n is an odd prime then f^(n-1)(n) = 2^(n-1) * n + 2^(n-1) - 1 is divisible by n since 2^(n-1) == 1 (mod n) by Fermat's theorem.
(End)

			a(2)=5 because 2, 5, 11, 23, 47 are prime but 95 is not.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A005384, A063378, A093008, A093007, A292936, A339579.
For records see A339581.
See also Cunningham chains, A005602, A005603.

Table[Length[NestWhileList[2#+1&,n,PrimeQ[#]&]],{n,100}]-1 (* Harvey P. Dale, Aug 08 2020 *)

a(n) = {if (! isprime(n), return (0)); d = 1; k = n; while(isprime(p = 2*k+1), k = p; d++;); return (d);} \\ Michel Marcus, Jul 22 2013

From Michael S. Branicky, Dec 24 2020: (Start)
See proof above.
a(n) = 0 if n == 0, 2, 4, 5, 6, 8 (mod 10), and n != 2 or 5.
a(n) <= 1 if n == 7 (mod 10).
a(n) <= 2 if n == 3 (mod 10).
a(n) <= 3 if n == 1 (mod 10).
(End)

Term a(1) = 0 prepended by Antti Karttunen, Oct 09 2018.

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

A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

Original entry on oeis.org

13, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 97, 101, 103, 109, 127, 137, 139, 149, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 257, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 439, 449

Offset: 1

Labos Elemer, Feb 05 2001

nonn

Primes which are neither safe nor of Sophie Germain type.
Primes not in Cunningham chains of the first kind. - Alonso del Arte, Jun 30 2005
A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

			Prime p=17 is here because both 35 and 8 are composite numbers. Such primes fall "out of" any Cunningham chain of first kind (or generate Cunningham chains of 0-length).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)
C. K. Caldwell, Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.

Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* Zak Seidov, Mar 09 2013 *)

is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ Charles R Greathouse IV, Jan 16 2013

a(n) ~ n log n. - Charles R Greathouse IV, Jan 16 2013

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

A109998 Non-Cunningham primes: primes isolated from any Cunningham chain under any iteration of 2p+-1 or (p+-1)/2.

Original entry on oeis.org

17, 43, 67, 71, 101, 103, 109, 127, 137, 149, 151, 163, 181, 197, 223, 241, 257, 269, 283, 311, 317, 349, 353, 373, 389, 401, 409, 433, 449, 461, 463, 487, 521, 523, 557, 569, 571, 599, 617, 631, 643, 647, 677, 701, 709, 739, 751, 769, 773, 787, 797, 821

Offset: 1

Alexandre Wajnberg, Sep 01 2005

The condition that neither 2p - 1 nor 2p + 1 be prime is equivalent to ((p-1) mod 3 = 0) or ((p+1) mod 3 = 0). For example, the prime p = 2^607 - 1 is not in this sequence because p + 1 mod 3 = 2. - Washington Bomfim, Oct 30 2009

			a(1) = 17 is here because 17 * 2 + 1 = 35, 17 * 2 - 1 = 33; (17+1)/2 = 9, (17-1)/2 = 8: four composite numbers.

Chris Caldwell's Prime Glossary, Cunningham chains.
Douglas S. Stones, On prime chains, arXiv:0908.2166 [math.NT] [From _Washington Bomfim_, Oct 30 2009, edited by _R. J. Mathar_, Mar 01 2010]

Cf. A005385, A005383, A060254, A005384, A005382, A068497, A059455, A059762, A057326, A023272, A023302, A059764, A057328, A023330, A005602, A064812, A005603.

nonCunninghamPrimes = {}; Do[p = Prime[n]; If[!PrimeQ[2p - 1] && !PrimeQ[2p + 1] && !PrimeQ[(p - 1)/2] && !PrimeQ[(p + 1)/2], AppendTo[nonCunninghamPrimes, p]], {n, 6!}]; nonCunninghamPrimes (* Vladimir Joseph Stephan Orlovsky, Mar 22 2009 *)

Corrected and extended by Ray Chandler, Sep 02 2005

Replaced link to cached arXiv URL with link to the abstract - R. J. Mathar, Mar 01 2010

A263879 Length k of the longest chain of primes p_1, p_2, ..., p_k such that p_1 is the n-th prime and p_{i+1} equals 2p_i + 1 or 2p_i - 1 for all i < k, the +/- sign depending on i.

Original entry on oeis.org

6, 5, 4, 2, 3, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 6, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 5, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1

Offset: 1

Jonathan Sondow, Oct 28 2015

nonn

If the +/- signs are all + or all -, then p_1, p_2, ..., p_k is a Cunningham chain of the first or second kind, respectively.
If p_1 > 3, then the +/- signs must be all + or all -, because if e = +1 or -1, then one of p, 2*p + e, 2*(2*p + e) - e is divisible by 3; see Löh (1989), p. 751.
Cunningham chains of the first and second kinds of length > 1 cannot begin with the same prime p > 3, because one of the numbers p, 2*p-1, 2*p+1 is divisible by 3.

			2, 3, 5, 11, 23, 47 is the longest such chain of primes starting with 2. Their indices are 1, 2, 3, 5, 9, 15, respectively, so a(1) = 6, a(2) = 5, a(3) = 4, a(5) = 3, a(9) = 2, and a(15) = 1.

R. K. Guy, Unsolved Problems in Number Theory, A7.

Robert Israel, Table of n, a(n) for n = 1..10000
G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Wikipedia, Cunningham chain

Cf. A005384, A005602, A181697, A181715.

f:= proc(n) option remember; local x;
  if n mod 3 = 1 then x:= 2*n-1 else x:= 2*n+1 fi;
  if isprime(x) then 1 + procname(x) else 1 fi;
end proc:
f(2):= 6: f(3):= 5:
map(f, [seq(ithprime(i),i=1..100)]); # Robert Israel, Jul 04 2023

A263879 = Join[{6, 5},
  Table[p = Prime[n]; cnt = 1;
   While[PrimeQ[2*p + 1] || PrimeQ[2*p - 1],
    cnt++ && If[PrimeQ[2*p + 1], p = 2*p + 1, p = 2*p - 1 ]];
   cnt, {n, 3, 100}]]

from sympy import prime, isprime
def A263879(n):
    if n <= 2: return 7-n
    p, c = prime(n), 1
    while isprime(p:=(p<<1)+(-1 if p%3==1 else 1)):
        c += 1
    return c # Chai Wah Wu, Jul 07 2023

a(n) = max(A181697(n), A181715(n)) for n > 2.

a(n) < prime(n) for n > 2; see Löh (1989), p. 751.

cp's OEIS Frontend

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

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A059456 Unsafe primes: primes not in A005385.

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A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

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A181697 Length of the complete Cunningham chain of the first kind starting with prime(n).

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A063377 Sophie Germain degree of n: number of iterations of n under f(k) = 2k+1 before we reach a number that is not a prime.

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

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A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

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A263879 Length k of the longest chain of primes p_1, p_2, ..., p_k such that p_1 is the n-th prime and p_{i+1} equals 2p_i + 1 or 2p_i - 1 for all i < k, the +/- sign depending on i.