A076336 -id:A076336 - cp's OEIS frontend

A270971 First differences of (provable) Sierpiński numbers (A076336).

192572, 448, 50946, 5216, 154980, 92322, 28672, 300270, 30926, 30522, 294348, 30898, 228104, 105316, 15362, 138154, 353430, 56, 60432, 318646, 31424, 34488, 355678, 224, 151732, 14336, 457534, 52658, 458752, 28856, 478140, 881790, 386158, 292716, 896, 422284, 119078, 1792, 63774

Offset: 1

Altug Alkan, Mar 27 2016

nonn

If we analyze the b-file of A076336, we see that most repeated values of a(n) are the form of 14*(2^k), for k >= 0. For the first 15000 (provable) Sierpiński numbers, there are 200 times 14, 201 times 28, 200 times 56, 200 times 112, 201 times 224, 200 times 448, 199 times 896, 200 times 1792, 200 times 3584 in this sequence.
Additionally, 14 appears as a minimum difference between consecutive (provable) Sierpiński numbers for the first 15000 terms that are listed in b-file of A076336. Graph of the sequence that is integers n such that A076336(n+1) = A076336(n) + 14 seems approximately linear.
The minimum value of this sequence is 2, as the numbers 3913004084027, 3913004084029 are consecutive odd numbers that are both Sierpinski numbers. - Robert Gelhar, Jul 23 2020

			a(1) = A076336(2) - A076336(1) = 271129 - 78557 = 192572.

Cf. A076336.

a(n) = A076336(n+1) - A076336(n).

A270993 Values of A076336(n) such that A076336(n) = A076336(n+1) - 14.

Original entry on oeis.org

7523267, 18708077, 29892887, 41077697, 52262507, 63447317, 74632127, 85816937, 97001747, 108186557, 119371367, 130556177, 141740987, 152925797, 164110607, 175295417, 186480227, 197665037, 208849847, 220034657, 231219467, 242404277, 253589087, 264773897, 275958707, 287143517, 298328327, 309513137

Offset: 1

Altug Alkan, Mar 28 2016

nonn

See A270971 for the motivation behind this sequence.
Riesel showed that there are infinitely many integers such that k*(2^m) - 1 is not prime for any integer m. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.
In this sequence, the lesser of (provable) Sierpiński pairs appears with the linear formula a(n) = 7523267 + 11184810*(n-1).
Since 7523267 is a term of A244561, for every integer k > 0, 7523267*2^k+1 has a divisor in the set {3, 5, 7, 13, 17, 241}. Because 11184810 = 2*3*5*7*13*17*241, a(n)*2^k+1 = 7523267*2^k+1 + 11184810*(n-1)*2^k+1 always has a divisor in the set {3, 5, 7, 13, 17, 241}. Since a(n) is always odd because of its definition, a(n) is a Sierpiński number. Additionally, 7523267 + 14 = 7523281 is also a term of A244561. So a(n) + 14 is a Sierpiński number too, with the same proof.
In conclusion, if the minimum difference between consecutive (provable) Sierpiński numbers is 14 (see comment section of A270971 for the reason behind this claim), a(n) and a(n) + 14 must be consecutive and a(n) = 7523267 + 11184810*(n-1) is the formula for this sequence.

			7523267 is a term because 7523267 and 7523267 + 14 = 7523281 are consecutive (provable) Sierpiński numbers.

Cf. A076336, A270971.

A271110 a(n) = A101036(n) - A076336(n).

Original entry on oeis.org

430646, 491572, 505572, 468318, 664338, 623962, 672132, 650628, 426224, 395410, 749622, 470874, 440004, 225336, 206090, 337014, 358670, 120306, 182388, 152680, 44666, 383260, 503380, 245786, 360250, 336066, 325314, 25308, 53278, -405460, -314318, -789560

Offset: 1

Altug Alkan, Mar 31 2016

sign

If we analyze the b-files of A076336 and A101036, we can see that the motivation of this sequence is oscillation similar graph of it. Since both sequences (A076336, A101036) contain the families of congruences, there are repeated values in this sequence. For the first 15000 terms, the most repeated values -376924 and -2843318 appear 28 times in this sequence.

			a(1) = A101036(1) - A076336(1) = 509203 - 78557 = 430646.

Cf. A076336, A101036.

A271111 Values of A076336(n) such that A076336(n) = A101036(n) + 376924.

Original entry on oeis.org

254695787, 265880597, 748135097, 758012237, 785868467, 792874337, 804059147, 806930417, 869860337, 893537627, 896408897, 949461677, 1501696307, 1556312687, 1567497497, 1602359597, 1647098837, 1668160787, 1697536277, 1698843947, 1757639267, 1826055797

Offset: 1

Altug Alkan, Mar 31 2016

nonn

Values of A076336(n) such that A076336(n) = A101036(n) + 2*2*17*23*241.
This sequence is an example for the relation between A076336 and A101036.
See A271110 for the motivation of "376924" that sequence focuses on.

			254695787 is a term because A076336(1714) = A101036(1714) + 2*2*17*23*241 = 254318863 + 2*2*17*23*241 = 254695787

Cf. A076336, A101036, A271110.

A271112 Values of A076336(n) such that A076336(n) = A101036(n) + 2843318.

Original entry on oeis.org

332252059, 341679859, 412107289, 479216149, 487082389, 530260069, 557509819, 568694629, 579879439, 586184119, 621300109, 1158170989, 1161489559, 1584950779, 1709545249, 1717411489, 1720730059, 1739781109, 1775092549, 1782958789, 1794143599, 1795705159

Offset: 1

Altug Alkan, Mar 31 2016

nonn

Values of A076336(n) such that A076336(n) = A101036(n) + 2*17*241*347.
This sequence is an example for the relation between A076336 and A101036.
See A271110 for the motivation of "2843318" that sequence focuses on.

			332252059 is a term because A076336(2208) = A101036(2208) + 2*17*241*347 = 329408741 + 2*17*241*347 = 332252059

Cf. A076336, A101036, A271110.

A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

Original entry on oeis.org

14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, 302, 304, 308, 314, 318

Offset: 1

N. J. A. Sloane

nonn

If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht, Dec 30 2005
Another subset of nontotients consists of the numbers j^2 + 1 such that j^2 + 2 is composite. These numbers j are given in A106571. Similarly, let b be 3 or a number such that b == 1 (mod 4). For any j > 0 such that b^j + 2 is composite, b^j + 1 is a nontotient. - T. D. Noe, Sep 13 2007
The Firoozbakht comment can be generalized: Observe that if k is a nontotient and 2k+1 is composite, then 2k is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^j is a nontotient for all j > 0. - T. D. Noe, Sep 13 2007

			There are no values of m such that phi(m)=14, so 14 is a term of the sequence.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 44 at p. 91.
R. K. Guy, Unsolved Problems in Number Theory, B36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 91.

Robert G. Wilson v, Table of n, a(n) for n = 1..29750 (terms 1..10000 from T. D. Noe).
Lambert A'Campo, Every 7-Dimensional Abelian Variety over the p-adic Numbers has a Reducible L-adic Galois Representation, arXiv:2006.06737 [math.NT], 2020.
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
K. Ford, S. Konyagin, and C. Pomerance, Residue classes free of values of Euler's function, arXiv:2005.01078 [math.NT] (1999).
L. Havelock, A Few Observations on Totient and Cototient Valence.
Eric Weisstein's World of Mathematics, Nontotient.
Wikipedia, Nontotient.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.
All even numbers not in A002202. Cf. A000010.
Cf. A005384, A006093, A014197, A057192, A076336, A079695, A106571.

a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
-- Reinhard Zumkeller, Nov 22 2015

[n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019

A005277 := n -> if type(n,even) and invphi(n)=[] then n fi: seq(A005277(i),i=1..318); # Peter Luschny, Jun 26 2011

searchMax = 320; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; Select[Range[searchMax], EvenQ[ # ] && (phiAnsYldList[[ # ]] == 0) &] (* Alonso del Arte, Sep 07 2004 *)
totientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)

is(n)=n%2==0 && !istotient(n) \\ Charles R Greathouse IV, Mar 04 2017

a(n) = 2*A079695(n). - R. J. Mathar, Sep 29 2021

{k: k even and A014197(k) = 0}. - R. J. Mathar, Sep 29 2021

More terms from Jud McCranie, Oct 13 2000

A101036 Riesel numbers n (n*2^k-1 is composite for all k>0, n odd) that have a covering set.

Original entry on oeis.org

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, 3079469, 3177553, 3292241, 3419789, 3423373, 3580901

Offset: 1

David W. Wilson, Jan 17 2005

nonn

Conjecture: there are infinitely many Riesel numbers that do not arise from a covering system. See page 16 of the Filaseta et al. reference. - Arkadiusz Wesolowski, Nov 17 2014
a(1) = 509203 is also the smallest odd n for which either n^p*2^k - 1 or abs(n^p - 2^k) is composite for every k > 0 and every prime p > 3. - Arkadiusz Wesolowski, Oct 12 2015
Theorem 11 of Filaseta et al. gives a Riesel number which is thought to violate the assumption of a periodic sequence of prime divisors mentioned in the title of this sequence. - Jeppe Stig Nielsen, Mar 16 2019
If the Riesel number mentioned in the previous comment does in fact not have a covering set, then this sequence is different from A076337, because then that number, 3896845303873881175159314620808887046066972469809^2, is a term of A076337, but not of this sequence. - Felix Fröhlich, Sep 09 2019
Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Jun 20 2021
Conjecture: if R is a Riesel number (that has a covering set), then there exists a prime P such that R^p is also a Riesel number for every prime p > P. - Thomas Ordowski, Jul 12 2022
Problem: are there numbers K such that K + 2^m is a Riesel number for every m > 0? If so, then (K + 2^m)*2^n - 1 is composite for every pair of positive integers m,n. Also, by the dual Riesel conjecture, |K + 2^m - 2^n| are always composite. Note that, by the dual Riesel conjecture, if p is an odd prime and n is a positive integer, then there exists n such that (p + 2^m)*2^n - 1 is prime. So if such a number K exists, it must be composite. - Thomas Ordowski, Jul 20 2022

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 238.

Pierre CAMI and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (P. CAMI supplied the first 335 terms)
Michael Filaseta, Carrie Finch and Mark Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.
Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.
Carrie E. Finch-Smith and R. Scottfield Groth, Arbitrarily Long Sequences of Sierpiński Numbers that are the Sum of a Sierpiński Number and a Mersenne Number, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.4. See p. 22.
Marcos J. González, Alberto Mendoza, Florian Luca, and V. Janitzio Mejía Huguet, On Composite Odd Numbers k for Which 2^n * k is a Noncototient for All Positive Integers n, J. Int. Seq., Vol. 24 (2021), Article 21.9.6.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
Wikipedia, Riesel number.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

See A076337 for references and additional information. Cf. A076336.

Up to 3292241, checked by Don Reble, Jan 17 2005, who comments that up to this point each n*2^k-1 has a prime factor <= 241.

New name from Felix Fröhlich, Sep 09 2019

A013597 a(n) = nextprime(2^n) - 2^n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 3, 1, 9, 7, 5, 3, 17, 27, 3, 1, 29, 3, 21, 7, 17, 15, 9, 43, 35, 15, 29, 3, 11, 3, 11, 15, 17, 25, 53, 31, 9, 7, 23, 15, 27, 15, 29, 7, 59, 15, 5, 21, 69, 55, 21, 21, 5, 159, 3, 81, 9, 69, 131, 33, 15, 135, 29, 13, 131, 9, 3, 33, 29, 25, 11, 15, 29

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

A013597 and A092131 use different definitions of "nextprime(2)", namely A151800 vs A007918: A013597 assumes nextprime(2) = 3 = A151800(2), whereas A092131 assumes nextprime(2) = 2 = A007918(n). [Edited by M. F. Hasler, Sep 09 2015]
If (for n>0) a(n)=1, then n is a power of 2 and 2^n+1 is a Fermat prime. n=1,2,4,8,16 are probably the only indices with this property. - Franz Vrabec, Sep 27 2005
Conjecture: there are no Sierpiński numbers in the sequence. See A076336. - Thomas Ordowski, Aug 13 2017

T. D. Noe, Table of n, a(n) for n = 0..5000
V. Danilov, Table for large n

Cf. A014210, A092131, A007918, A151800.

A013597 := proc(n)
    nextprime(2^n)-2^n ;
end proc:
seq(A013597(n),n=0..40) ;

Table[NextPrime[#] - # &[2^n], {n, 0, 73}] (* Michael De Vlieger, Aug 15 2017 *)

a(n) = nextprime(2^n+1) - 2^n; \\ Michel Marcus, Nov 06 2015

from sympy import nextprime
def A013597(n): return nextprime(m:=1<Chai Wah Wu, Dec 02 2024

a(n) = A151800(2^n) - 2^n = A013632(2^n). - R. J. Mathar, Nov 28 2016

Conjecture: a(n) < n^2/2 for n > 1. - Thomas Ordowski, Aug 13 2017

A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite.

Original entry on oeis.org

509203

Offset: 1

N. J. A. Sloane, Nov 07 2002

509203 has been proved to be a member of the sequence, and is conjectured to be the smallest member. However, as of 2009, there are still several smaller numbers which are candidates and have not yet been ruled out (see links).
Riesel numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k-1 and disproved by finding prime n*2^k-1. It is conjectured that numbers that cannot be proved Riesel in this way are non-Riesel. However, some numbers resist both proof and disproof.
Others conjecture the opposite: that there are infinitely many Riesel numbers that do not arise from a covering system, see A101036. The word "odd" is needed in the definition because otherwise for any term n, all numbers n*2^m, m >= 1, would also be Riesel numbers, but we don't want them in this sequence (as is manifest from A101036). Since 1 and 3 obviously are not in this sequence, for any n in this sequence n-1 is an even number > 2 and therefore composite, so one could replace "k >= 1" equivalently by "k >= 0". - M. F. Hasler, Aug 20 2020
Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Apr 02 2022

R. K. Guy, Unsolved Problems in Number Theory, Section B21.
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed., 1989, p. 282.

Ray Ballinger and Wilfrid Keller, The Riesel Problem: Definition and Status [http://www.prothsearch.com/rieselprob.html].
Chris Caldwell, Riesel Numbers.
Chris Caldwell, Sierpinski Numbers.
Yves Gallot, A search for some small Brier numbers, 2000.
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.8.
Tanya Khovanova, Non Recursions.
Joe McLean, Brier Numbers.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Riesel numbers.
Index entries for one-term sequences.

Main sequences for Riesel problem: A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A076336, A076335, A003261.

Normally we require at least four terms but we will make an exception for this sequence in view of its importance. - N. J. A. Sloane, Nov 07 2002. See A101036 for the most likely extension.
Edited by N. J. A. Sloane, Nov 13 2009
Definition corrected ("odd" added) by M. F. Hasler, Aug 23 2020

A076335 Brier numbers: numbers that are both Riesel and Sierpiński [Sierpinski], or odd n such that for all k >= 1 the numbers n2^k + 1 and n2^k - 1 are composite.

Original entry on oeis.org

3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, 21444598169181578466233, 28960674973436106391349, 32099522445515872473461, 32904995562220857573541

Offset: 1

Olivier Gérard, Nov 07 2002

nonn

a(1), a(4), and a(6)-a(8) computed by Christophe Clavier, Dec 31 2013 (see link below). 10439679896374780276373 had been found earlier in 2013 by Dan Ismailescu and Peter Seho Park (see reference below). a(3), a(5), and a(9) computed in 2014 by Emmanuel Vantieghem.
These are just the smallest examples known - there may be smaller ones.
There are no Brier numbers below 10^9. - Arkadiusz Wesolowski, Aug 03 2009
Other Brier numbers are 143665583045350793098657, 1547374756499590486317191, 3127894363368981760543181, 3780564951798029783879299, but these may not be the /next/ Brier numbers after those shown. From 2002 to 2013 these four numbers were given here as the smallest known Brier numbers, so the new entry A234594 has been created to preserve that fact. - N. J. A. Sloane, Jan 03 2014
143665583045350793098657 computed in 2007 by Michael Filaseta, Carrie Finch, and Mark Kozek.
It is a conjecture that every such number has more than 10 digits. In 2011 I have calculated that for any n < 10^10 there is a k such that either n*2^k + 1 or n*2^k - 1 has all its prime factors greater than 1321. - Arkadiusz Wesolowski, Feb 03 2016 [Editor's note: The comment below states that the conjecture is now proved. - M. F. Hasler, Oct 06 2021]
There are no Brier numbers below 10^10.  For each n < 10^10, there exists at least one prime of the form n*2^k-1 or n*2^k+1 with k <= 356981. The largest necessary prime is 1355477231*2^356981+1. - Kellen Shenton, Oct 25 2020

D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, and M. Kozek, Polygonal, Sierpinski, and Riesel numbers, Journal of Integer Sequences, 2015 Vol 18. #15.8.1.
Chris Caldwell, The Prime Glossary, Riesel number
Chris Caldwell, The Prime Glossary, Sierpinski number
Christophe Clavier, 14 new Brier numbers
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.
P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.
M. Filaseta et al., On Powers Associated with Sierpiński Numbers, Riesel Numbers and Polignac’s Conjecture, Journal of Number Theory, Volume 128, Issue 7, July 2008, Pages 1916-1940. (See pages 9-10)
Michael Filaseta and Jacob Juillerat, Consecutive primes which are widely digitally delicate, arXiv:2101.08898 [math.NT], 2021.
Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022. See also Integers (2023) Vol. 23, #A75.
Yves Gallot, A search for some small Brier numbers, 2000.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 6992565235279559197457863
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.
Joe McLean, Brier Numbers [Cached copy]
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 58. Brier numbers revisited, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 68. More on Brier numbers, The Prime Puzzles and Problems Connection.
Carlos Rivera, See here for latest information about progress on this sequence
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261, A364412, A364413.
A180247 gives the primes.
See also A076336, A076337.
A234594 is the old, incorrect version.

Many terms reported in Problem 29 from "The Prime Problems & Puzzles Connection" from Carlos Rivera, May 30 2010
Entry revised by Arkadiusz Wesolowski, May 17 2012
Entry revised by Carlos Rivera and N. J. A. Sloane, Jan 03 2014
Entry revised by Arkadiusz Wesolowski, Feb 15 2014

cp's OEIS Frontend

A270971 First differences of (provable) Sierpiński numbers (A076336).

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A270993 Values of A076336(n) such that A076336(n) = A076336(n+1) - 14.

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A271110 a(n) = A101036(n) - A076336(n).

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A271111 Values of A076336(n) such that A076336(n) = A101036(n) + 376924.

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A271112 Values of A076336(n) such that A076336(n) = A101036(n) + 2843318.

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A005277 Nontotients: even numbers k such that phi(m) = k has no solution.

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A101036 Riesel numbers n (n*2^k-1 is composite for all k>0, n odd) that have a covering set.

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A013597 a(n) = nextprime(2^n) - 2^n.

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A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite.

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A076335 Brier numbers: numbers that are both Riesel and Sierpiński [Sierpinski], or odd n such that for all k >= 1 the numbers n2^k + 1 and n2^k - 1 are composite.