A364413 -id:A364413 - cp's OEIS frontend

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.

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

A222534 Smallest Sierpinski number that is divisible by the n-th prime.

Original entry on oeis.org

7592506760633776533, 36293948155, 157957457, 603713, 422590909, 78557, 6134663, 1259779, 575041, 7892569, 2931991, 4095859, 2541601, 7892569, 29169451, 271577, 35193889, 12824269, 603713, 9454157, 575041, 7696009, 5455789, 41561687, 7400371, 2191531, 29046541, 2931991

Offset: 2

Arkadiusz Wesolowski, Feb 24 2013

nonn

For an odd prime p and odd k, if p divides k, then p does not divide k*2^n + 1 for any n.

			603713 is first Sierpinski number that is divisible by 11, the 5th prime - so a(5) = 603713.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary: Sierpinski number
Carlos Rivera, Problem 86. A smaller Sierpinski integer divided by 3, The Prime Puzzles & Problems Connection.

Cf. A076336, A187716, A213529, A364413.

Offset changed and initial term added by Arkadiusz Wesolowski, May 11 2017

a(2) corrected by Arkadiusz Wesolowski, Jul 27 2023

cp's OEIS Frontend

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.

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A222534 Smallest Sierpinski number that is divisible by the n-th prime.

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A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A222534 Smallest Sierpinski number that is divisible by the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

Views

Author

Keywords

Links

Crossrefs

Formula

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.