A064699 -id:A064699 - 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

A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p2^k + 1 and p2^k - 1 are composite.

Original entry on oeis.org

10439679896374780276373, 21444598169181578466233, 105404490005793363299729, 178328409866851219182953, 239365215362656954573813, 378418904967987321998467, 422280395899865397194393, 474362792344501650476113, 490393518369132405769309

Offset: 1

Arkadiusz Wesolowski, Aug 19 2010

nonn

WARNING: These are just the smallest examples known - there may be smaller ones. Even the first term is uncertain. - N. J. A. Sloane, Jun 20 2017
There are no prime Brier numbers below 10^10. - Arkadiusz Wesolowski, Jan 12 2011
It is a conjecture that every such number has more than 11 digits. In 2011 I have calculated that for any prime p < 10^11 there is a k such that either p*2^k + 1 or p*2^k - 1 has all its prime factors greater than 1321. - Arkadiusz Wesolowski, Feb 03 2016
The first term was found by Dan Ismailescu and Peter Seho Park and the next two by Christophe Clavier (see below). See also A076335. - N. J. A. Sloane, Jan 03 2014
a(4)-a(9) computed in 2017 by the author.

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.
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 52. ±p ± 2^n, The Prime Puzzles and Problems Connection.
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.

These are the primes in A076335.

Entry revised by N. J. A. Sloane, Jan 03 2014

Entry revised by Arkadiusz Wesolowski, May 29 2017

A194635 Indices of records in A194591 restricted to prime indices.

Original entry on oeis.org

2, 5, 13, 47, 59, 109, 241, 631, 1109, 1373, 1447, 16229, 52267, 56543, 838441, 16935761, 270704167, 3296757029

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

Integers for which the smallest k in A194591 such that prime(n)*2^k - 1 or prime(n)*2^k + 1 is prime (A194608) increases.
a(19) > 10^10.
A194607 gives the record values of A194606.

Wilfrid Keller, personal communication, 2010.

Chris K. Caldwell, The List of Largest Known Primes, 270704167 * 2^85461 - 1
Chris K. Caldwell, The List of Largest Known Primes, 3296757029 * 2^191207 + 1
Carlos Rivera, Problem 30. A stepladder to B-alpha sequence, The Prime Puzzles & Problems Connection.

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194636, A194637, A194638, A194639.

Cf. A064699, A076335, A180247, A217892.

l = -1; Flatten[Table[p = Prime[n]; k = 0; While[! PrimeQ[p*2^k - 1] && ! PrimeQ[p*2^k + 1], k++]; If[k > l, l = k; p, {}], {n, 10^4}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

a(17) was found in 2000 by Wilfrid Keller
a(18) was found in 2003 by Patrick De Geest
Edited by Max Alekseyev, Oct 14 2012
Edited by Arkadiusz Wesolowski, Sep 11 2013

A194639 Indices of records in A194591 when it is restricted to odd indices.

Original entry on oeis.org

1, 5, 13, 47, 59, 109, 241, 335, 1109, 1373, 1447, 14893, 52267, 56543, 649603, 838441, 8840101, 16935761, 100604513, 118373279, 270704167, 1355477231

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

Integers for which the smallest k in A194591 such that (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1 is prime (A194638) increases.

A194637 gives the record values of A194636.

Wilfrid Keller, personal communication, 2010.

Chris K. Caldwell, The List of Largest Known Primes, 1355477231 * 2^356981 + 1
Carlos Rivera, Problem 30

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638.
Cf. A064699, A076335, A180247, A217892.
Cf. A217892 (indices of records of unrestricted A194591)

l = -1; Flatten[Table[n = 2*n - 1; k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; If[k > l, l = k; n, {}], {n, 10^5}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

a(22) was found in 2002 by Wilfrid Keller.

Definition corrected by Max Alekseyev and Farideh Firoozbakht, Oct 16 2014

A103964 Record values in A040076.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165

Offset: 1

Lei Zhou, Feb 24 2005

The index sequence of this sequence is 1, 3, 7, 17, 19, 31, 47, 383, 2897. A040076(2897)>8192, not yet found.

A064699 gives where the records occur.

			A040076(1)=0, so a(1)=0;
A040076(3)=1, so a(2)=1;

Cf. A040076, A103963.

k = -1; n = 0; km = k; While[k < 8192, n++; k = 0; cp = n*(2^ k) + 1; While[(! PrimeQ[cp]) && (k < 8192), k++; cp = n*(2^k) + 1]; If[k > km, km = k; Print[{n, km}]]]

Extended by T. D. Noe, Nov 15 2010

a(13) was found by PrimeGrid, added by Richard N. Smith, Jul 15 2019

A350118 Primes p for which the smallest m such that p*2^m + 1 is prime increases. Sequence terminates with the smallest prime Sierpiński number.

Original entry on oeis.org

2, 3, 7, 17, 19, 31, 47, 383, 2897, 3061, 5297, 7013, 10223

Offset: 1

Jianing Song, Dec 14 2021

The smallest prime Sierpiński number is likely to be 271129.
Related to A058887: this sequence is A058887 with repeated values removed. The following list shows that relation between these two sequences:
a(2) = 3, A350119(2) = 1 => A058887(0..0) = 3;
a(3) = 7, A350119(3) = 2 => A058887(1..1) = 7;
a(4) = 17, A350119(4) = 3 => A058887(2..2) = 17;
a(5) = 19, A350119(5) = 6 => A058887(3..5) = 19;
a(6) = 31, A350119(6) = 8 => A058887(6..7) = 31;
a(7) = 47, A350119(7) = 583 => A058887(8..582) = 47;
a(8) = 383, A350119(8) = 6393 => A058887(583..6392) = 383;
...
a(N) is the smallest prime Sierpiński number, A350119(N) = -1 => A058887(k) = a(N) for all k >= A350119(N-1).

			Let b(p) be the smallest m such that p*2^m + 1 is prime. We have a(1) = 2 with b(2) = 0.
The least prime p such that b(p) > 0 is p = 3 with b(3) = 1, so a(2) = 3.
The least prime p such that b(p) > 1 is p = 7 with b(7) = 2, so a(3) = 7.
The least prime p such that b(p) > 2 is p = 17 with b(17) = 3, so a(4) = 17.
The least prime p such that b(p) > 3 is p = 19 with b(19) = 6, so a(5) = 19.
The least prime p such that b(p) > 6 is p = 31 with b(31) = 8, so a(6) = 31.
The least prime p such that b(p) > 8 is p = 47 with b(47) = 583, so a(7) = 47.

Cf. A058887, A057192, A350119, A064699, A076336 (Sierpiński numbers).

b(p) = for(k=0, oo, if(isprime(p*2^k+1), return(k)))
list(lim) = if(lim>=2, my(v=[2],r=0); forprime(p=2, lim, if(b(p)>r, r=b(p); v=concat(v,p))); v)

A258074 Table read by rows: each row represents the constant and exponent of a Colbert number.

Original entry on oeis.org

5359, 5054502, 33661, 7031232, 28433, 7830457, 27653, 9167433, 19249, 13018586, 10223, 31172165

Offset: 1

Tom Edgar, May 18 2015

Each row has two entries [k,n]. With this notation, the corresponding Colbert number is k*2^n+1.
A Colbert number is an integer with more than 1,000,000 digits that is prime and has contributed to the in-progress computational proof that 78557 is the smallest Sierpiński number (A076336).
a(11)-a(12) number, corresponding to [10223,31172165], is the second largest known prime that is not a Mersenne prime as of August 2023. - Hermann Stamm-Wilbrandt, Aug 13 2023
This table can only have (and is expected to have) five more rows corresponding to constants k equal to 21181, 22699, 24737, 55459, and 67607.

			The table is as follows:
  5359, 5054502;
  33661, 7031232;
  28433, 7830457;
  27653, 9167433;
  19249, 13018586
  10223, 31172165

C. K. Caldwell, The Prime Glossary, Colbert Number
C. K. Caldwell, The Prime Database, 10223*2^31172165+1
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Hermann Stamm-Wilbrandt, Colbert numbers, contains entries [k,n,s,x,y] for the 6 Colbert numbers, with p=k*2^n+1, s^2%p==p-1 and p==x^2+y^2.
Wikipedia, Seventeen or Bust

Cf. A076336, A064699, A064721.

a(11)-a(12) from Richard N. Smith, Jul 15 2019, following by the prime 10223*2^31172165+1 found by PrimeGrid.

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

A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p*2^k + 1 and p*2^k - 1 are composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A194635 Indices of records in A194591 restricted to prime indices.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A194639 Indices of records in A194591 when it is restricted to odd indices.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A103964 Record values in A040076.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A350118 Primes p for which the smallest m such that p*2^m + 1 is prime increases. Sequence terminates with the smallest prime Sierpiński number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A258074 Table read by rows: each row represents the constant and exponent of a Colbert number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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.

A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p2^k + 1 and p2^k - 1 are composite.