A101036 -id:A101036 - cp's OEIS frontend

A271110 a(n) = A101036(n) - A076336(n).

430646, 491572, 505572, 468318, 664338, 623962, 672132, 650628, 426224, 395410, 749622, 470874, 440004, 225336, 206090, 337014, 358670, 120306, 182388, 152680, 44666, 383260, 503380, 245786, 360250, 336066, 325314, 25308, 53278, -405460, -314318, -789560

Offset: 1

Altug Alkan, Mar 31 2016

sign

If we analyze the b-files of A076336 and A101036, we can see that the motivation of this sequence is oscillation similar graph of it. Since both sequences (A076336, A101036) contain the families of congruences, there are repeated values in this sequence. For the first 15000 terms, the most repeated values -376924 and -2843318 appear 28 times in this sequence.

			a(1) = A101036(1) - A076336(1) = 509203 - 78557 = 430646.

Cf. A076336, A101036.

A271111 Values of A076336(n) such that A076336(n) = A101036(n) + 376924.

Original entry on oeis.org

254695787, 265880597, 748135097, 758012237, 785868467, 792874337, 804059147, 806930417, 869860337, 893537627, 896408897, 949461677, 1501696307, 1556312687, 1567497497, 1602359597, 1647098837, 1668160787, 1697536277, 1698843947, 1757639267, 1826055797

Offset: 1

Altug Alkan, Mar 31 2016

nonn

Values of A076336(n) such that A076336(n) = A101036(n) + 2*2*17*23*241.
This sequence is an example for the relation between A076336 and A101036.
See A271110 for the motivation of "376924" that sequence focuses on.

			254695787 is a term because A076336(1714) = A101036(1714) + 2*2*17*23*241 = 254318863 + 2*2*17*23*241 = 254695787

Cf. A076336, A101036, A271110.

A271112 Values of A076336(n) such that A076336(n) = A101036(n) + 2843318.

Original entry on oeis.org

332252059, 341679859, 412107289, 479216149, 487082389, 530260069, 557509819, 568694629, 579879439, 586184119, 621300109, 1158170989, 1161489559, 1584950779, 1709545249, 1717411489, 1720730059, 1739781109, 1775092549, 1782958789, 1794143599, 1795705159

Offset: 1

Altug Alkan, Mar 31 2016

nonn

Values of A076336(n) such that A076336(n) = A101036(n) + 2*17*241*347.
This sequence is an example for the relation between A076336 and A101036.
See A271110 for the motivation of "2843318" that sequence focuses on.

			332252059 is a term because A076336(2208) = A101036(2208) + 2*17*241*347 = 329408741 + 2*17*241*347 = 332252059

Cf. A076336, A101036, A271110.

A345685 a(n) is the smallest cardinality of all covering sets associated with Riesel number A101036(n).

Original entry on oeis.org

6, 6, 7, 7, 6, 7, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6

Offset: 1

Felix Fröhlich, Jun 23 2021

The condition for choosing a covering set is necessary as there are Riesel numbers with more than one covering set, see A263392.

			   n | Riesel number | Covering set               | a(n)
--------------------------------------------------------
   1 |  509203       | {3, 5, 7, 13, 17, 241}     | 6
   2 |  762701       | {3, 5, 7, 13, 17, 241}     | 6
   3 |  777149       | {3, 5, 7, 13, 19, 37, 73}  | 7
   4 |  790841       | {3, 5, 7, 13, 19, 37, 73}  | 7
   5 |  992077       | {3, 5, 7, 13, 17, 241}     | 6
   6 | 1106681       | {3, 5, 7, 13, 19, 37, 73}  | 7
   7 | 1247173       | {3, 5, 7, 13, 17, 241}     | 6
   8 | 1254341       | {3, 5, 7, 13, 17, 241}     | 6
   9 | 1330207       | {3, 5, 7, 13, 17, 241}     | 6
  10 | 1330319       | {3, 5, 7, 13, 17, 241}     | 6
  11 | 1715053       | {3, 5, 7, 13, 19, 37, 73}  | 7
  12 | 1730653       | {3, 5, 7, 13, 17, 241}     | 6
  13 | 1730681       | {3, 5, 7, 13, 17, 241}     | 6
  14 | 1744117       | {3, 5, 7, 13, 19, 73, 109} | 7
  15 | 1830187       | {3, 5, 7, 13, 37, 73, 109} | 7
  16 | 1976473       | {3, 5, 7, 13, 17, 241}     | 6

Prime Wiki, Category:Riesel k=Riesel [Links to pages giving the covering sets of Riesel numbers]
Wikipedia, Riesel number

Cf. A101036, A244070, A244071, A244072, A244073, A244074, A263392.

A076336 (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.

Original entry on oeis.org

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, 3083723, 3098059, 3555593, 3608251

Offset: 1

N. J. A. Sloane, Nov 07 2002

"Provable" in the definition means provable by any method (whether using a covering set or not). - N. J. A. Sloane, Aug 03 2024
It is only a conjecture that this sequence is complete up to 3000000 - there may be missing terms.
It is conjectured that 78557 is the smallest Sierpiński number. - T. D. Noe, Oct 31 2003
Sierpiński numbers may be proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k+1 and disproved by finding a prime n*2^k+1. It is conjectured by some people that numbers that cannot be proved to be Sierpiński by this method are non-Sierpiński. However, some numbers resist both proof and disproof. - David W. Wilson, Jan 17 2005 [Edited by N. J. A. Sloane, Aug 03 2024]
Sierpiński showed that this sequence is infinite.
There are four related sequences that arise in this context:
S1: Numbers n such that n*2^k + 1 is composite for all k (this sequence)
S2: Odd numbers n such that 2^k + n is composite for all k (apparently it is conjectured that S1 and S2 are the same sequence)
S3: Numbers n such that n*2^k + 1 is prime for all k (empty)
S4: Numbers n such that 2^k + n is prime for all k (empty)
The following argument, due to Michael Reid, attempts to show that S3 and S4 are empty: If p is a prime divisor of n + 1, then for k = p - 1, the term (either n*2^k + 1 or 2^k + n) is a multiple of p (and also > p, so not prime). [However, David McAfferty points that for the case S3, this argument fails if p is of the form 2^m-1. So it may only be a conjecture that the set S3 is empty. - N. J. A. Sloane, Jun 27 2021]
a(1) = 78557 is also the smallest odd n for which either n^p*2^k + 1 or n^p + 2^k is composite for every k > 0 and every prime p greater than 3. - Arkadiusz Wesolowski, Oct 12 2015
n = 4008735125781478102999926000625 = (A213353(1))^4 is in this sequence but is thought not to satisfy the conjecture mentioned by David W. Wilson above. For this multiplier, all n*2^(4m + 2) + 1 are composite by an Aurifeuillean factorization. Only the remaining cases, n*2^k + 1 where k is not 2 modulo 4, are covered by a finite set of primes (namely {3, 17, 97, 241, 257, 673}). See Izotov link for details (although with another prime set). - Jeppe Stig Nielsen, Apr 14 2018
Conjecture: if S is a (provable) Sierpiński number, then there exists a prime P such that S^p is also a Sierpiński number for every prime p > P. - Thomas Ordowski, Jul 12 2022
Problem: are there odd numbers K such that K - 2^m is a Sierpiński number for every 1 < 2^m < K? If so, then all positive values of (K - 2^m)*2^n + 1 are composite. Also, by the dual Sierpiński conjecture, K - 2^m + 2^n is composite for every 1 < 2^m < K and for every n > 0. Note that, by the dual Sierpiński conjecture, if p is an odd prime and 1 < 2^m < p, 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
From M. F. Hasler, Jul 23 2022: (Start)
1) The above Conjecture is true for Sierpiński numbers provable by a "covering set", with P equal to the largest prime factor of the elements of that set*, according to the explanation from Michael Filaseta posted Jul 12 2022 on the SeqFan mailing list, cf. links. (*More generally: for S^p with any p coprime to all elements of the covering set, but not necessarily prime.)
2) Wilson's comment from 2005 (also the first part, not only the conjecture) is misleading if not wrong because there are provable Sierpiński numbers for which a covering set is not known (maybe even believed not to exist), as explained by Nielsen in his above comment from 2018. (End)

R. K. Guy, Unsolved Problems in Number Theory, Section B21.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 420.
Paulo Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 237-238.

T. D. Noe and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (T. D. Noe supplied 13394 terms which came from McLean. a(1064), a(7053), and a(13397)-a(15000) from Arkadiusz Wesolowski.)
Chris Caldwell, Riesel number
Chris Caldwell, Sierpinski number
Michael Filaseta, quoted by T. Ordowski, Re: Is it true? SeqFan mailing list, Jul 12 2022
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.
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, 16 (2013), #13.9.8.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.
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.
J. McLean, Searching for large Sierpinski numbers [Cached copy]
J. McLean, Brier Numbers [Cached copy]
Don Reble, Proofs concerning S3 and S4
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Payam Samidoost, Dual Sierpinski problem search page [Broken link?]
Payam Samidoost, Dual Sierpinski problem search page [Cached copy]
Payam Samidoost, 4847 [Broken link?]
Payam Samidoost, 4847 [Cached copy]
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Seventeen or Bust, A Distributed Attack on the Sierpinski Problem
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]
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind
Wikipedia, Sierpiński number.

Cf. A003261, A052333, A076335, A076337, A101036, A137715, A263169, A305473, A306151.

A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.

Original entry on oeis.org

1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 6804302445, 7525056375, 7602256605, 9055691835, 9217432215

Offset: 1

Charles R Greathouse IV, Feb 13 2009

Crocker shows that this sequence is infinite.
All members above 5 found so far (up to 2.5 * 10^11) are divisible by 255 = 3 * 5 * 17, and many are divisible by 257. I conjecture that all members of this sequence greater than 5 are divisible by 255. This implies that all odd numbers (greater than 7) are the sum of a prime and at most three positive powers of two.
Pan shows that, for every c > 1, a(n) << x^c. More specifically, there are constants C,D > 0 such that there are at least Dx/exp(C log x log log log log x/log log log x) members of this sequence up to x. - Charles R Greathouse IV, Apr 11 2016
All terms > 5 are numbers k > 3 such that k - 2^n is a de Polignac number (A006285) for every n > 0 with 2^n < k. Are there numbers K such that |K - 2^n| is a Riesel number (A101036) for every n > 0? If so, ||K - 2^n| - 2^m| is composite for every pair m,n > 0, by the dual Riesel conjecture. - Thomas Ordowski, Jan 06 2024
In keeping with the example's connection to A000215, the lowest ki for ki * Product_{i=0..11} (F(i)) to belong to A156695 are 1, 433007, 25471, 17047, 1291, 7, 101, 807, 83, 347, 9, 179. So for example, 433007*(3*5) is a term. This implies a variant of the first commented conjecture accordingly. - Bill McEachen, Apr 17 2025

			Prime factorization of terms:
F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257 are Fermat numbers (cf. A000215)
6495105    = 3   * 5   * 17               * 25471
848629545  = 3   * 5   * 17               * 461      * 7219
1117175145 = 3   * 5   * 17         * 257 * 17047
2544265305 = 3^2 * 5   * 17         * 257 * 12941
3147056235 = 3^2 * 5   * 17         * 257 * 16007
3366991695 = 3   * 5   * 17   * 83  * 257 * 619
3472109835 = 3   * 5   * 17         * 257 * 52981
3621922845 = 3   * 5   * 17^2       * 257 * 3251
3861518805 = 3^3 * 5   * 17         * 257 * 6547
4447794915 = 3^3 * 5   * 17         * 257 * 7541
4848148485 = 3^4 * 5   * 17               * 704161
5415281745 = 3   * 5   * 17               * 21236399
5693877405 = 3^2 * 5   * 17         * 257 * 28961
6804302445 = 3^2 * 5   * 17   * 53  * 257 * 653
7525056375 = 3^2 * 5^3 * 17         * 257 * 1531
7602256605 = 3   * 5   * 17         * 257 * 311      * 373
9055691835 = 3   * 5   * 17         * 257 * 138181
9217432215 = 3^2 * 5   * 17   * 173 * 257 * 271

Giovanni Resta, Table of n, a(n) for n = 1..233 (terms < 10^12)
Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.
Roger Crocker, Some counter-examples in the additive theory of numbers, Master's thesis (Ohio State University), 1962.
Hao Pan, On the integers not of the form p + 2^a + 2^b. arXiv:0905.3809 [math.NT], 2009.
Zhi-Wei Sun, Mixed sums of primes and other terms (2009-2010).

Cf. A006285, A118955, A232565, A337487.

is(n)=if(n%2==0,return(0)); for(a=1,log(n)\log(2), for(b=1,a, if(isprime(n-2^a-2^b),return(0)))); 1 \\ Charles R Greathouse IV, Nov 27 2013

from itertools import count, islice
from sympy import isprime
def A156695_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+(startvalue&1^1),1),2):
        l = n.bit_length()-1
        for a in range(l,0,-1):
            c = n-(1<A156695_list = list(islice(A156695_gen(),4)) # Chai Wah Wu, Nov 29 2023

Factorizations added by Daniel Forgues, Jan 20 2011

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

cp's OEIS Frontend

A271110 a(n) = A101036(n) - A076336(n).

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A271111 Values of A076336(n) such that A076336(n) = A101036(n) + 376924.

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A271112 Values of A076336(n) such that A076336(n) = A101036(n) + 2843318.

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A345685 a(n) is the smallest cardinality of all covering sets associated with Riesel number A101036(n).

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A076336 (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite.

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A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.

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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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