A076335 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.

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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..9.

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

Crossrefs

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.

Sequence in context: A257376 A037017 A337094 * A280301 A115542 A343266

Adjacent sequences: A076332 A076333 A076334 * A076336 A076337 A076338

Keyword

nonn

Author

Olivier Gérard, Nov 07 2002

Extensions

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

Status

approved