

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.


26



3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, 21444598169181578466233, 28960674973436106391349, 32099522445515872473461, 32904995562220857573541
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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 a new entry has been created to preserve that fact, A234594.  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


REFERENCES

D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, M. Kozek, Polygonal, Sierpinski, and Riesel numbers, Journal of Integer, 2015 Vol 18. #15.8.1.


LINKS

Table of n, a(n) for n=1..9.
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. 7981.
P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113123.
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 19161940. (See pages 910)
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
Carlos Rivera, Problem 58
Carlos Rivera, See here for latest information about progress on this sequence
Eric Weisstein's World of Mathematics, Brier Number


MATHEMATICA

lst1 = {}; lst2 = {}; u = {{1, 3}, {2, 5}, {6, 7}, {7, 11}, {11, 13}, {8, 17}, {10, 19}, {23, 31}, {4, 37}, {45, 61}, {41, 73}, {48, 97}, {105, 109}, {33, 151}, {233, 241}, {129, 257}, {2, 331}, {16, 1321}}; p = Times @@ Take[u, All, 1]; q = Flatten[u]; Do[d = p/q[[2*a]]; r = Reduce[d*x == q[[2*a  1]], x, Modulus > q[[2*a]]]; If[Length[r] > 0, AppendTo[lst1, d*Last[r]], Abort[]], {a, Length[q]/2}]; c = FromDigits[p]; i = FromDigits@Total[lst1]; n = 0; While[True, i = NestWhile[#/2 &, Abs[i + (1)^n*c], EvenQ]; n++; If[MemberQ[lst2, i], Print@First@Sort[lst2]; Break[]]; If[n == 360, Break[]]; AppendTo[lst2, i]] (* Arkadiusz Wesolowski, Feb 12 2013 *)


CROSSREFS

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261.
A180247 gives the primes.
See also A076336, A076337.
A234594 is the old, incorrect, version.
Sequence in context: A257376 A037017 A187716 * A280301 A115542 A171265
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



