|
%I #134 Oct 04 2023 12:07:36
%S 3316923598096294713661,10439679896374780276373,
%T 11615103277955704975673,12607110588854501953787,
%U 17855036657007596110949,21444598169181578466233,28960674973436106391349,32099522445515872473461,32904995562220857573541
%N Brier numbers: numbers that are both Riesel and Sierpiński [Sierpinski], or odd n such that for all k >= 1 the numbers n*2^k + 1 and n*2^k - 1 are composite.
%C 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_.
%C These are just the smallest examples known - there may be smaller ones.
%C There are no Brier numbers below 10^9. - _Arkadiusz Wesolowski_, Aug 03 2009
%C 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
%C 143665583045350793098657 computed in 2007 by Michael Filaseta, Carrie Finch, and Mark Kozek.
%C 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]
%C 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
%H D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, and M. Kozek, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.html">Polygonal, Sierpinski, and Riesel numbers</a>, Journal of Integer Sequences, 2015 Vol 18. #15.8.1.
%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/RieselNumber.html">Riesel number</a>
%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/SierpinskiNumber.html">Sierpinski number</a>
%H Christophe Clavier, <a href="/A076335/a076335.txt">14 new Brier numbers</a>
%H Fred Cohen and J. L. Selfridge, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0376583-0">Not every number is the sum or difference of two prime powers</a>, Math. Comput. 29 (1975), pp. 79-81.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1950-07.pdf">On integers of the form 2^k + p and some related problems</a>, Summa Brasil. Math. 2 (1950), pp. 113-123.
%H M. Filaseta et al., <a href="http://www.math.sc.edu/~filaseta/papers/SierpinskiEtCoPapNew.pdf">On Powers Associated with Sierpiński Numbers, Riesel Numbers and Polignac’s Conjecture</a>, Journal of Number Theory, Volume 128, Issue 7, July 2008, Pages 1916-1940. (See pages 9-10)
%H Michael Filaseta and Jacob Juillerat, <a href="https://arxiv.org/abs/2101.08898">Consecutive primes which are widely digitally delicate</a>, arXiv:2101.08898 [math.NT], 2021.
%H Michael Filaseta, Jacob Juillerat, and Thomas Luckner, <a href="https://arxiv.org/abs/2209.10646">Consecutive primes which are widely digitally delicate and Brier numbers</a>, arXiv:2209.10646 [math.NT], 2022. See also <a href="http://math.colgate.edu/~integers/x75/x75.pdf">Integers</a> (2023) Vol. 23, #A75.
%H Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf">A search for some small Brier numbers</a>, 2000.
%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/cpage/23122.html">Prime Curios! 6992565235279559197457863</a>
%H Dan Ismailescu and Peter Seho Park, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.html">On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences</a>, Journal of Integer Sequences, 16 (2013), #13.9.8.
%H Joe McLean, <a href="http://oeis.org/A076336/a076336b.html">Brier Numbers</a> [Cached copy]
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_029.htm">Problem 29. Brier numbers</a>, The Prime Puzzles and Problems Connection.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_058.htm">Problem 58. Brier numbers revisited</a>, The Prime Puzzles and Problems Connection.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_068.htm">Problem 68. More on Brier numbers</a>, The Prime Puzzles and Problems Connection.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/private/index.htm">See here for latest information about progress on this sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BrierNumber.html">Brier Number</a>
%Y 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.
%Y A180247 gives the primes.
%Y See also A076336, A076337.
%Y A234594 is the old, incorrect version.
%K nonn
%O 1,1
%A _Olivier Gérard_, Nov 07 2002
%E Many terms reported in Problem 29 from "The Prime Problems & Puzzles Connection" from _Carlos Rivera_, May 30 2010
%E Entry revised by _Arkadiusz Wesolowski_, May 17 2012
%E Entry revised by _Carlos Rivera_ and _N. J. A. Sloane_, Jan 03 2014
%E Entry revised by _Arkadiusz Wesolowski_, Feb 15 2014
| 