A050921 -id:A050921 - cp's OEIS frontend

A064699 Integers for which the smallest m in A040076 such that n*2^m+1 is prime (A050921) increases.

1, 3, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223

Offset: 1

Robert G. Wilson v, Oct 16 2001

Where records occur in A040076. A103964 gives the record values.

a = -1; Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; If[m > a, a = m; Print[n]], {n, 1, 10^3} ]

Corrected and extended by T. D. Noe, Nov 15 2010

a(13) was found by PrimeGrid, added by Arkadiusz Wesolowski, Feb 13 2017

A064721 Potential Sierpiński numbers: integers for which the smallest m > 2^10 in A040076 such that n*2^m+1 is prime (A050921).

Original entry on oeis.org

383, 766, 881, 1532, 1643, 1762, 2897, 3061, 3064, 3286, 3443, 3524, 3829, 4847, 4861, 5297, 5359, 5794, 5897, 6122, 6128, 6319, 6572, 6886, 7013, 7352, 7493, 7651, 7658, 7909, 7957, 8119, 8269, 8423, 8543, 8929, 9323, 9694, 9722

Offset: 1

Robert G. Wilson v, Oct 16 2001

nonn

The first confirmed Sierpiński number is 78557.

Cf. A040076, A050921.

Do[m = 0; While[m <= 2^10 && !PrimeQ[n*2^m + 1], m++ ]; If[m > 2^10, Print[n]], {n, 1, 10^4} ]

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.

Original entry on oeis.org

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

A040076 Smallest m >= 0 such that n*2^m + 1 is prime, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, 4, 3, 0, 2, 3, 1, 0, 1, 2, 4, 1, 2, 0, 1, 1, 8, 7, 2, 582, 1, 0, 2, 1, 1, 0, 3, 0

Offset: 1

David W. Wilson

Sierpiński showed that a(n) = -1 infinitely often. John Selfridge showed that a(78557) = -1 and it is conjectured that a(n) >= 0 for all n < 78557.

Determining a(131072) = a(2^17) is equivalent to finding the next Fermat prime after F_4 = 2^16 + 1. - Jeppe Stig Nielsen, Jul 27 2019

			1*(2^0)+1=2 is prime, so a(1)=0;
3*(2^1)+1=5 is prime, so a(3)=1;
For n=7, 7+1 and 7*2+1 are composite, but 7*2^2+1=29 is prime, so a(7)=2.

T. D. Noe, Table of n, a(n) for n = 1..10000 (with help from the Sierpiński problem website)
Ray Ballinger and Wilfrid Keller, The Sierpiński Problem: Definition and Status
Seventeen or Bust, A Distributed Attack on the Sierpiński problem

For the corresponding primes see A050921.
Cf. A103964, A040081.
Cf. A033809, A046067 (odd n), A057192 (prime n).

Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; Print[m], {n, 1, 110} ]
sm[n_]:=Module[{k=0},While[!PrimeQ[n 2^k+1],k++];k]; Array[sm,120] (* Harvey P. Dale, Feb 05 2020 *)

A038699 Riesel problem: Smallest prime of form n*2^m-1, m >= 0, or 0 if no such prime exists.

Original entry on oeis.org

3, 3, 2, 3, 19, 5, 13, 7, 17, 19, 43, 11, 103, 13, 29, 31, 67, 17, 37, 19, 41, 43, 367, 23, 199, 103, 53, 223, 463, 29, 61, 31, 131, 67, 139, 71, 73, 37, 311, 79, 163, 41, 5503, 43, 89, 367, 751, 47, 97, 199, 101, 103, 211, 53, 109, 223, 113, 463, 241663, 59, 487, 61

Offset: 1

N. J. A. Sloane, Dec 30 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..650
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.

Primes arising in A040081 (or 0 if no prime exists).
Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.
Cf. A050921, A010051, A000079.

a038699 = until ((== 1) . a010051) ((+ 1) . (* 2)) . (subtract 1)
-- Reinhard Zumkeller, Mar 05 2012

getm[n_]:=Module[{m=0},While[!PrimeQ[n 2^m-1],m++];n 2^m-1]; Array[getm,80]  (* Harvey P. Dale, Apr 24 2011 *)

More terms from Henry Bottomley, Apr 24 2001

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

A194603 Smallest prime either of the form n2^k - 1 or n2^k + 1, k >= 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 2, 3, 11, 5, 13, 7, 17, 11, 23, 11, 53, 13, 29, 17, 67, 17, 37, 19, 41, 23, 47, 23, 101, 53, 53, 29, 59, 29, 61, 31, 67, 67, 71, 37, 73, 37, 79, 41, 83, 41, 173, 43, 89, 47, 751, 47, 97, 101, 101, 53, 107, 53, 109, 113, 113, 59, 1889, 59, 487, 61, 127

Offset: 1

Arkadiusz Wesolowski, Aug 30 2011

nonn

Primes arising from A194591 (or 0 if no such prime exists).

Many of these terms are in A093868.

			For n=7, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(7)=13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A194591, A194600, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639.

Cf. A038699, A050921, A076335, A180247.

Table[k = 0; While[! PrimeQ[a = n*2^k - 1] && ! PrimeQ[a = n*2^k + 1], k++]; a, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)
n2k[n_]:=Module[{k=0},While[NoneTrue[n*2^k+{1,-1},PrimeQ],k++];SelectFirst[ n*2^k+{-1,1},PrimeQ]]; Array[n2k,70] (* The program uses the NoneTrue and SelectFirst functions from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2015 *)

A194608 Smallest prime either of the form prime(n)2^k - 1 or prime(n)2^k + 1, k >= 0, or 0 if no such prime exists, where prime(n) denotes the n-th prime number.

Original entry on oeis.org

3, 2, 11, 13, 23, 53, 67, 37, 47, 59, 61, 73, 83, 173, 751, 107, 1889, 487, 269, 283, 293, 157, 167, 179, 193, 809, 823, 857, 6977, 227, 509, 263, 547, 277, 1193, 2417, 313, 653, 2671, 347, 359, 1447, 383, 773, 787, 397, 421, 1783, 907, 457, 467, 479, 493567

Offset: 1

Arkadiusz Wesolowski, Aug 30 2011

nonn

Primes arising from A194606 (or 0 if no such prime exists).

			For n=4, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(4)=13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A194591, A194600, A194603, A194606, A194607, A194635, A194636, A194637, A194638, A194639.

Cf. A038699, A050921, A076335, A180247.

Table[p = Prime[n]; k = 0; While[! PrimeQ[a = p*2^k - 1] && ! PrimeQ[a = p*2^k + 1], k++]; a, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)

A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 11, 13, 17, 23, 53, 29, 67, 37, 41, 47, 101, 53, 59, 61, 67, 71, 73, 79, 83, 173, 89, 751, 97, 101, 107, 109, 113, 1889, 487, 127, 131, 269, 137, 283, 293, 149, 307, 157, 163, 167, 1361, 173, 179, 181, 373, 191, 193, 197, 809, 823, 211, 857, 6977, 223, 227

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

nonn

Bisection of A194603.

Primes arising from A194636 (or 0 if no such prime exists).

			For n=4, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(4)=13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194639.

Cf. A038699, A050921, A076335, A180247.

Table[n = 2*n - 1; k = 0; While[! PrimeQ[a = n*2^k - 1] && ! PrimeQ[a = n*2^k + 1], k++]; a, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)

A225721 Starting with x = n, the number of iterations of x := 2x - 1 until x is prime, or -1 if no prime exists.

Original entry on oeis.org

-1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, 4, 3, 0, 2, 3, 1, 0, 1, 2, 4

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

sign

This appears to be a shifted variant of A040076. - R. J. Mathar, May 28 2013

If n is prime, then a(n) = 0. If the sequence never reaches a prime number (for n = 1) or the prime number has more than 1000 digits, -1 is used instead. There are 22 such numbers for n < 10000.

			For a(20), the trajectory is 20->39->77->153->305->609->1217, a prime number. That required 6 steps, so a(20)=6.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A050921 (primes obtained).
Cf. A040081, A038699, A050412, A052333, A046069 (related to the Riesel problem).
Cf. A000668, A000043, A065341 (Mersenne primes), A000079 (powers of 2).
Cf. A007770 (happy numbers), A031177 (unhappy numbers).
Cf. A037274 (home primes), A037271 (steps), A037272, A037272.

y=as.bigz(rep(0,500)); ys=rep(0,500);
for(i in 1:500) { n=as.bigz(i); k=0;
    while(isprime(n)==0 & ndig(n)<1000 & k<5000) { k=k+1; n=2*n-1 }
    if(ndig(n)>=1000 | k>=5000) { ys[i]=-1; y[i]=-1;
    } else {ys[i]=k; y[i]=n; }
}

cp's OEIS Frontend

A064699 Integers for which the smallest m in A040076 such that n*2^m+1 is prime (A050921) increases.

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A064721 Potential Sierpiński numbers: integers for which the smallest m > 2^10 in A040076 such that n*2^m+1 is prime (A050921).

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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.

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A040076 Smallest m >= 0 such that n*2^m + 1 is prime, or -1 if no such m exists.

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A038699 Riesel problem: Smallest prime of form n*2^m-1, m >= 0, or 0 if no such prime exists.

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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.

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A194603 Smallest prime either of the form n*2^k - 1 or n*2^k + 1, k >= 0, or 0 if no such prime exists.

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A194608 Smallest prime either of the form prime(n)*2^k - 1 or prime(n)*2^k + 1, k >= 0, or 0 if no such prime exists, where prime(n) denotes the n-th prime number.

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A194638 Smallest prime either of the form (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1, k >= 0, or 0 if no such prime exists.

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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.

A194603 Smallest prime either of the form n2^k - 1 or n2^k + 1, k >= 0, or 0 if no such prime exists.

A194608 Smallest prime either of the form prime(n)2^k - 1 or prime(n)2^k + 1, k >= 0, or 0 if no such prime exists, where prime(n) denotes the n-th prime number.

A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.