A040076 -id:A040076 - cp's OEIS frontend

A103964 Record values in A040076.

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

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

Original entry on oeis.org

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.

Cf. A040076, A050921.

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

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

A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, 2, 1, 1, 4, 2, 5, 4, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 4, 2, 1, 8, 2, 1, 2, 1, 3, 16, 1, 3, 6, 1, 1, 2, 3, 1, 8, 6, 1, 2, 3, 1, 4, 1, 3, 2, 1, 53, 6, 8, 3, 4, 1, 1, 8, 6, 3, 2, 1, 7, 2, 8, 1, 2, 2, 1, 4, 1, 3, 6, 1, 1, 2, 4, 15, 2

Offset: 1

Eric W. Weisstein

sign

There exist odd integers 2k-1 such that (2k-1)2^n+1 is always composite.
The smallest known example is 78557. Therefore a(39279) = -1.
For the corresponding primes see A057025(n-1), n >= 1, where a 0 will show up if a(n) = -1. - Wolfdieter Lang, Feb 07 2013.
Jaeschke shows that every positive integer appears infinitely often. - Jeppe Stig Nielsen, Jul 06 2020

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.

T. D. Noe, Table of n, a(n) for n = 1..5000 (with help from the Sierpiński problem website; typo in a(3707)=1 corrected by Jeppe Stig Nielsen)
Ray Ballinger and Wilfrid Keller, Sierpiński Problem
John R. Cowles and Ruben Gamboa, Verifying Sierpiński and Riesel Numbers in ACL2, arXiv preprint arXiv:1110.4671 [cs.DM], 2011.
G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.
Seventeen or Bust, A Distributed Attack on the Sierpiński Problem
W. Sierpiński, Sur un problème concernant les nombres k*2^n+1, Elem. d. Math. 15, pp. 73-74, 1960.
Eric Weisstein's World of Mathematics, Riesel Number.
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind.

Cf. A046068.
Bisection of A040076. Cf. A033809.
Cf. A057192, A057025.

max = 10000 (* this maximum value of m is sufficient up to n = 1000 *); a[n_] := For[m = 1, m <= max, m++, If[PrimeQ[(2n - 1)*2^m + 1], Return[m]]] /. Null -> -1; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 08 2012 *)

A194591 Least k >= 0 such that n2^k - 1 or n2^k + 1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 5, 0, 3, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 1

Offset: 1

Arkadiusz Wesolowski, Aug 29 2011

sign

Fred Cohen and J. L. Selfridge showed that a(n) = -1 infinitely often.

a(n) = 0 iff n is in A045718.

			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)=1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79-81.
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639.
Cf. A040081, A040076, A076335, A180247.
Cf. A217892 and A194600 (indices and values of the records).

Table[k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; k, {n, 100}] (* T. D. Noe, Aug 29 2011 *)

If a(n)>0, then a(2n)=a(n)-1.

A194606 Least k >= 0 such that prime(n)2^k - 1 or prime(n)2^k + 1 is prime, or -1 if no such value exists, where prime(n) denotes the n-th prime number.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 1, 5, 3, 2, 2, 2, 1, 1, 1, 1, 3, 3, 3, 6, 1, 2, 1, 2, 1, 3, 4, 1, 2, 4, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 11, 1, 4, 2, 3, 1, 2, 1, 11, 1, 1, 9, 3, 6, 1, 1, 3, 3, 4, 1, 1, 2, 1, 2, 11, 4, 3, 2, 1, 4, 1, 2, 1, 1

Offset: 1

Arkadiusz Wesolowski, Aug 30 2011

sign

A194607 gives the record values.

			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)=1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194591, A194600, A194603, A194607, A194608, A194635, A194636, A194637, A194638, A194639.

Cf. A040081, A040076, A076335, A180247.

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

A194636 Least k >= 0 such that (2n-1)2^k - 1 or (2n-1)2^k + 1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 5, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 3, 6, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 5, 1, 3, 4, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

sign

Bisection of A194591: a(n) = A194591(2*n-1).

A194637 gives the record values.

			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)=1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194637, A194638, A194639.

Cf. A040081, A040076, A076335, A180247.

Table[n = 2*n - 1; k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; k, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)
p[n_]:=Module[{c=2n-1,k=0},While[!Or@@PrimeQ[c*2^k+{1,-1}],k++];k]; Array[ p,90] (* Harvey P. Dale, Mar 08 2013 *)

A050921 Smallest prime of form n*2^m+1, m >= 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 137, 19, 1217, 41, 43, 23, 47, 97, 101, 53, 109, 29, 59, 31, 7937, 257, 67, 137, 71, 37, 149, 1217, 79, 41, 83, 43, 173, 89, 181, 47

Offset: 1

N. J. A. Sloane, Dec 30 1999

Primes arising from A040076 (or 0 if no such prime exists).

Or: Starting with x=n+1, the first prime created by iterating the map x-> 2*x-1. - Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

R. J. Mathar, Table of n, a(n) for n = 1..382

Cf. A040076, A034694, A038699.

A050921 := proc(n)
    for m from 0 do
        if isprime(n*2^m+1) then
            return n*2^m+1 ;
        end if;
    end do;
end proc; # R. J. Mathar, Jun 01 2013

Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; Print[n*2^m + 1], {n, 1, 47} ]

The next term (47*2^583 + 1) is too large to show.

cp's OEIS Frontend

A103964 Record values in A040076.

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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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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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A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists.

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A194591 Least k >= 0 such that n*2^k - 1 or n*2^k + 1 is prime, or -1 if no such value exists.

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A194606 Least k >= 0 such that prime(n)*2^k - 1 or prime(n)*2^k + 1 is prime, or -1 if no such value exists, where prime(n) denotes the n-th prime number.

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A194636 Least k >= 0 such that (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1 is prime, or -1 if no such value 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.

A194591 Least k >= 0 such that n2^k - 1 or n2^k + 1 is prime, or -1 if no such value exists.

A194606 Least k >= 0 such that prime(n)2^k - 1 or prime(n)2^k + 1 is prime, or -1 if no such value exists, where prime(n) denotes the n-th prime number.

A194636 Least k >= 0 such that (2n-1)2^k - 1 or (2n-1)2^k + 1 is prime, or -1 if no such value exists.