A108129 -id:A108129 - cp's OEIS frontend

A108130 Partial sums of A108129.

2, 3, 5, 6, 7, 9, 12, 13, 15, 16, 17, 21, 24, 25, 29, 30, 32, 34, 35, 38, 40, 47, 48, 52, 53, 54, 56, 57, 58, 70, 73, 75, 79, 84, 85, 87, 94, 95, 97, 98, 101, 103, 108, 109, 113, 114, 117, 119, 120, 121, 131, 134, 136, 146, 155, 157, 165, 166, 167, 179, 180

Offset: 1

Jorge Coveiro, Jun 04 2005

nonn

Cf. A108129.

f(n) = my(k=2*n-1, m=1); while(!isprime(k*2^m-1), m++); m; \\ A108129
a(n) = sum(k=1, n, a(k));

a(14) and following corrected by Georg Fischer, Dec 26 2022

Offset 1 from Michel Marcus, Dec 26 2022

A101036 Riesel numbers n (n*2^k-1 is composite for all k>0, n odd) that have a covering set.

Original entry on oeis.org

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, 3079469, 3177553, 3292241, 3419789, 3423373, 3580901

Offset: 1

David W. Wilson, Jan 17 2005

nonn

Conjecture: there are infinitely many Riesel numbers that do not arise from a covering system. See page 16 of the Filaseta et al. reference. - Arkadiusz Wesolowski, Nov 17 2014
a(1) = 509203 is also the smallest odd n for which either n^p*2^k - 1 or abs(n^p - 2^k) is composite for every k > 0 and every prime p > 3. - Arkadiusz Wesolowski, Oct 12 2015
Theorem 11 of Filaseta et al. gives a Riesel number which is thought to violate the assumption of a periodic sequence of prime divisors mentioned in the title of this sequence. - Jeppe Stig Nielsen, Mar 16 2019
If the Riesel number mentioned in the previous comment does in fact not have a covering set, then this sequence is different from A076337, because then that number, 3896845303873881175159314620808887046066972469809^2, is a term of A076337, but not of this sequence. - Felix Fröhlich, Sep 09 2019
Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Jun 20 2021
Conjecture: if R is a Riesel number (that has a covering set), then there exists a prime P such that R^p is also a Riesel number for every prime p > P. - Thomas Ordowski, Jul 12 2022
Problem: are there numbers K such that K + 2^m is a Riesel number for every m > 0? If so, then (K + 2^m)*2^n - 1 is composite for every pair of positive integers m,n. Also, by the dual Riesel conjecture, |K + 2^m - 2^n| are always composite. Note that, by the dual Riesel conjecture, if p is an odd prime and n is a positive integer, then there exists n such that (p + 2^m)*2^n - 1 is prime. So if such a number K exists, it must be composite. - Thomas Ordowski, Jul 20 2022

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 238.

Pierre CAMI and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (P. CAMI supplied the first 335 terms)
Michael Filaseta, Carrie Finch and Mark Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.
Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.
Carrie E. Finch-Smith and R. Scottfield Groth, Arbitrarily Long Sequences of Sierpiński Numbers that are the Sum of a Sierpiński Number and a Mersenne Number, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.4. See p. 22.
Marcos J. González, Alberto Mendoza, Florian Luca, and V. Janitzio Mejía Huguet, On Composite Odd Numbers k for Which 2^n * k is a Noncototient for All Positive Integers n, J. Int. Seq., Vol. 24 (2021), Article 21.9.6.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
Wikipedia, Riesel number.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

See A076337 for references and additional information. Cf. A076336.

Up to 3292241, checked by Don Reble, Jan 17 2005, who comments that up to this point each n*2^k-1 has a prime factor <= 241.

New name from Felix Fröhlich, Sep 09 2019

A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite.

Original entry on oeis.org

509203

Offset: 1

N. J. A. Sloane, Nov 07 2002

509203 has been proved to be a member of the sequence, and is conjectured to be the smallest member. However, as of 2009, there are still several smaller numbers which are candidates and have not yet been ruled out (see links).
Riesel numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k-1 and disproved by finding prime n*2^k-1. It is conjectured that numbers that cannot be proved Riesel in this way are non-Riesel. However, some numbers resist both proof and disproof.
Others conjecture the opposite: that there are infinitely many Riesel numbers that do not arise from a covering system, see A101036. The word "odd" is needed in the definition because otherwise for any term n, all numbers n*2^m, m >= 1, would also be Riesel numbers, but we don't want them in this sequence (as is manifest from A101036). Since 1 and 3 obviously are not in this sequence, for any n in this sequence n-1 is an even number > 2 and therefore composite, so one could replace "k >= 1" equivalently by "k >= 0". - M. F. Hasler, Aug 20 2020
Named after the Swedish mathematician Hans Ivar Riesel (1929-2014). - Amiram Eldar, Apr 02 2022

R. K. Guy, Unsolved Problems in Number Theory, Section B21.
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed., 1989, p. 282.

Ray Ballinger and Wilfrid Keller, The Riesel Problem: Definition and Status [http://www.prothsearch.com/rieselprob.html].
Chris Caldwell, Riesel Numbers.
Chris Caldwell, Sierpinski Numbers.
Yves Gallot, A search for some small Brier numbers, 2000.
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.9.8.
Tanya Khovanova, Non Recursions.
Joe McLean, Brier Numbers.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Riesel numbers.
Index entries for one-term sequences.

Main sequences for Riesel problem: A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A076336, A076335, A003261.

Normally we require at least four terms but we will make an exception for this sequence in view of its importance. - N. J. A. Sloane, Nov 07 2002. See A101036 for the most likely extension.
Edited by N. J. A. Sloane, Nov 13 2009
Definition corrected ("odd" added) by M. F. Hasler, Aug 23 2020

A050412 Riesel problem: start with n; repeatedly double and add 1 until reaching a prime. Sequence gives number of steps to reach a prime or 0 if no prime is ever reached.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 4, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 3, 2, 1, 3, 4, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 7, 24, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 12, 2, 3, 4, 2, 1, 4, 1, 5, 2, 1, 1, 2, 4, 7, 2552, 1, 1, 2, 2, 1, 4, 3, 1, 2, 1, 5, 6, 1, 23, 4, 1, 1, 2, 3, 3, 2, 1, 1, 4, 1, 1

Offset: 1

Robert G. Wilson v, Dec 22 1999

a(n) is the smallest m >= 1 such that (n+1)*2^m - 1 is prime (or 0 if no such prime exists).
It is conjectured that n = 509203 is the smallest Riesel number, i.e., n*2^k - 1 is composite for every k>0. - Robert G. Wilson v, Mar 01 2015. [This would imply that a(509203) is the first zero term in this sequence. - N. J. A. Sloane, Jul 31 2024]
Comment from N. J. A. Sloane, Aug 01 2024 (Start)
Both the Ballinger-Keller and Prime Wiki links assert that 104917*2^340181-1 is prime, but leave open the possibility that there is an m < 340181 which makes 104917*2^m - 1 a prime.
This question was finally settled by Lucas A. Brown on Jul 31 2024, who showed that m = 340181 is the smallest value that gives a prime. This implies that a(104917) = 340181.
Brown used a Python program (see below), with BPSW for the primality testing and gmpy2 to handle the arithmetic. The program was started on Jul 30 2024 and finished on Jul 31 2024.
He reports that it took about 15 hours in wall-clock time, and used 24 threads running in parallel.  (End)

			For n=4; the smallest m>=1 such that (4+1)*2^m-1 is prime is m=2: 5*2^2-1=19 (prime). - _Jaroslav Krizek_, Feb 13 2011

Robert G. Wilson v, Table of n, a(n) for n = 1..2291 (first 657 terms from T. D. Noe)
Ray Ballinger and Wilfrid Keller, The Riesel Problem: Definition and Status, Proth Search Page.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
Prime Wiki, Riesel primes of the form 104917•2^n-1

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A051914, A052333 (primes reached), A052334, A052339, A052340, A050413, A078680, A101036.

A050412 := proc(n)
    local twox1,k ;
    twox1 := 2*n+1 ;
    k := 1;
    while not isprime(twox1) do
        twox1 := 2*twox1+1 ;
        k := k+1 ;
    end do:
    return k;
end proc: # R. J. Mathar, Jul 23 2015

a[n_] := Block[{s=n, c=1}, While[ ! PrimeQ[2*s+1], s = 2*s+1; c++]; c]; Table[ a[n], {n, 1, 99} ] (* Jean-François Alcover, Feb 06 2012, after Pari *)
a[n_] := Block[{k = 1}, While[ !PrimeQ[2^k (n + 1) - 1], k++];k]; Array[a, 100] (* Robert G. Wilson v, Feb 14 2015 *) (* Corrected by Paolo Xausa, Jul 30 2024 *)

a(n)=if(n<0,0,s=n; c=1; while(isprime(2*s+1)==0,s=2*s+1; c++); c)
(Python, designed specifically for n = 104917)
#! /usr/bin/env python3
from labmath import primegen, isprime, mpz, count
from multiprocessing import Pool
primes = list(primegen(1000000))
def test(n):
    for p in primes:
        if (104917 * pow(2, n, p)) % p == 1:
            return (n, False)
    return (n, isprime(104917 * mpz(2)**n - 1, tb=[]))
with Pool(24) as P:
    for (n, result) in P.imap(test, count()):
        print('\b'*80, n, end='', flush=True)
        if result:
            break # Lucas A. Brown, Aug 01 2024

If a(n) = k with k>1, then a(2n+1) = k-1. - Robert G. Wilson v, Mar 01 2015

If a(n) = 0, then a(2n+1) is also 0. Conjecture: If a(n) = 1, then a(2n+1) is not 0. - Jeppe Stig Nielsen, Feb 12 2023

More terms from Christian G. Bower, Dec 23 1999

Second definition corrected by Jaroslav Krizek, Feb 13 2011

A040081 Riesel problem: a(n) = smallest m >= 0 such that n*2^m-1 is prime, or -1 if no such prime exists.

Original entry on oeis.org

2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 4, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 7, 0, 1, 3, 4, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 3, 12, 0, 3, 0, 2, 1, 4, 1, 5, 0, 1, 1, 2, 0, 7, 0, 1, 1, 2, 2, 1, 0, 3, 1, 2, 0, 5, 6, 1, 23, 4, 0, 1, 2, 3, 3, 2, 1, 1, 0, 1, 1, 10, 0, 3

Offset: 1

David W. Wilson

nonn

Eric Chen, Table of n, a(n) for n = 1..2292 (first 1000 terms from T. D. Noe)
Matteo Cati and Dmitrii V. Pasechnik, A database of constructions of Hadamard matrices, arXiv:2411.18897 [math.CO], 2024. See p. 14.
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A010051, A000079.

a040081 = length . takeWhile ((== 0) . a010051) .
                       iterate  ((+ 1) . (* 2)) . (subtract 1)
-- Reinhard Zumkeller, Mar 05 2012

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

a(n)=for(k=0,2^16,if(ispseudoprime(n*2^k-1), return(k))) \\ Eric Chen, Jun 01 2015

from sympy import isprime
def a(n):
  m = 0
  while not isprime(n*2**m - 1): m += 1
  return m
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 01 2021

A052333 Riesel problem: start with n; repeatedly double and add 1 until reach a prime. Sequence gives a(n) = prime reached, or 0 if no prime is ever reached.

Original entry on oeis.org

3, 5, 7, 19, 11, 13, 31, 17, 19, 43, 23, 103, 223, 29, 31, 67, 71, 37, 79, 41, 43, 367, 47, 199, 103, 53, 223, 463, 59, 61, 127, 131, 67, 139, 71, 73, 151, 311, 79, 163, 83, 5503, 738197503, 89, 367, 751, 191, 97, 199, 101, 103, 211, 107, 109, 223, 113, 463

Offset: 1

Christian G. Bower, Dec 19 1999

Equivalently, a(n) = smallest prime of form (n+1)*2^k-1 for k >= 1, or 0 if no such prime exists.
a(509202) = 0 (i.e. never reaches a prime) - Chris Nash (chris_nash(AT)hotmail.com). (Of course this is related to the entry 509203 of A076337.)
a(73) is a 771-digit prime reached after 2552 iterations - Warut Roonguthai. This was proved to be a prime by Paul Jobling (Paul.Jobling(AT)WhiteCross.com) using PrimeForm and by Ignacio Larrosa Cañestro using Titanix (http://www.znz.freesurf.fr/pages/titanix.html). [Oct 30 2000]

			a(4)=19 because 4 -> 9 (composite) -> 19 (prime).

N. J. A. Sloane, Table of n, a(n) for n = 1..657. Note that A050412(658) = 800516, so a(658) has 240983 digits, which is too large for a b-file.
Ray Ballinger and Wilfrid Keller, The Riesel Problem: Definition and Status
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

CMain sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A051914, A052334, A052339, A052340, A040081.

Table[NestWhile[2#+1&,2n+1,!PrimeQ[#]&,1,1000],{n,60}] (* Harvey P. Dale, May 08 2011 *)

a(n)=while(!isprime(n=2*n+1),);n \\ oo loop when a(n) = 0. - Charles R Greathouse IV, May 08 2011

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

A046069 Riesel Problem: Smallest m >= 0 such that (2n-1)2^m-1 is prime, or -1 if no such value exists.

Original entry on oeis.org

2, 0, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, 1, 4, 1, 1, 2, 1, 1, 12, 3, 2, 4, 5, 1, 2, 7, 1, 2, 1, 3, 2, 5, 1, 4, 1, 3, 2, 1, 1, 10, 3, 2, 10, 9, 2, 8, 1, 1, 12, 1, 2, 2, 25, 1, 2, 3, 1, 2, 1, 1, 2, 5, 1, 4, 5, 3, 2, 1, 1, 2, 3, 2, 4, 1, 2, 2, 1, 1, 8, 3, 4, 2, 1, 3, 226, 3, 1, 2

Offset: 1

Eric W. Weisstein

nonn

There exist odd integers 2k-1 such that (2k-1)2^n-1 is always composite.

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..1000
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.
Eric Weisstein's World of Mathematics, Riesel Number.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.
Cf. A046067, A046070.
Bisection of A040081.

max = 10^6; (* this maximum value of m is sufficient up to n=1000 *) a[1] = 2; a[2] = 0; a[n_] := For[m = 1, m <= max, m++, If[PrimeQ[(2*n - 1)*2^m - 1], Return[m]]] /. Null -> -1; Reap[ Do[ Print[ "a(", n, ") = ", a[n]]; Sow[a[n]], {n, 1, 100}]][[2, 1]] (* Jean-François Alcover, Nov 15 2013 *)

A250205 Riesel problem in base 6: Least k > 0 such that n*6^k-1 is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 2, 2, 0, 4, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 3, 0, 1, 1, 6, 2, 0, 5, 1, 1, 1, 0, 6, 2, 1, 1, 0, 1, 2, 10, 1, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 1, 8, 1, 1, 0, 1, 2, 2, 4, 0, 49, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 6, 2, 0, 1, 1, 1, 1, 0, 5, 1, 1, 2, 0, 1, 10, 2, 1

Offset: 1

Eric Chen, Mar 13 2015

nonn

a(5j+1) = 0 except for a(1), since (5j+1)*6^k-1 is always divisible by 5, but there are infinitely many numbers not in the form 5j+1 such that a(n) = 0.
a(n) = 0 for n == 84687 mod 10124569, because then n*6^k-1 is always divisible by at least one of 7, 13, 31, 37, 97. - Robert Israel, Mar 17 2015
Conjecture: if n is not in the form 5j+1 and n < 84687, then a(n) > 0.

Eric Chen, Table of n, a(n) for n = 1..1000
Gary Barnes, Riesel conjectures and proofs

Cf. A040081, A046069, A050412, A108129.

Cf. A250204 (Least k > 0 such that n*6^k+1 is prime).

N:= 1000: # to get a(1) to a(N), using k up to 10000
a[1]:= 1:
for n from 2 to N do
if n mod 5 = 1 then a[n]:= 0
else
    for k from 1 to 10000 do
    if isprime(n*6^k-1) then
       a[n]:= k;
         break
      fi
    od
fi
od:
seq(a[n],n=1..N); # Robert Israel, Mar 17 2015

(* m <= 10000 is sufficient up to n = 1000 *)
a[n_] := For[k = 1, k <= 10000, k++, If[PrimeQ[n*6^k - 1], Return[k]]] /. Null -> 0; Table[a[n], {n, 1, 120}]

a(n) = if(n%5==1 && n>1, 0, for(k = 1, 10000, if(ispseudoprime(n*6^k-1), return(k))))

a(A024898(n)) = 1. - Michel Marcus, Mar 16 2015

cp's OEIS Frontend

A108130 Partial sums of A108129.

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A101036 Riesel numbers n (n*2^k-1 is composite for all k>0, n odd) that have a covering set.

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A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite.

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A050412 Riesel problem: start with n; repeatedly double and add 1 until reaching a prime. Sequence gives number of steps to reach a prime or 0 if no prime is ever reached.

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A040081 Riesel problem: a(n) = smallest m >= 0 such that n*2^m-1 is prime, or -1 if no such prime exists.

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A052333 Riesel problem: start with n; repeatedly double and add 1 until reach a prime. Sequence gives a(n) = prime reached, or 0 if no prime is ever reached.

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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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A046069 Riesel Problem: Smallest m >= 0 such that (2n-1)2^m-1 is prime, or -1 if no such value exists.

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