A101036 -id:A101036 - cp's OEIS frontend

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

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 *)

A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

Original entry on oeis.org

2, 1, 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, 1, 1, 2

Offset: 1

Jorge Coveiro, Jun 04 2005

nonn

It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
Browkin & Schinzel, having proved that 509203*2^k - 1 is composite for all k > 0, ask for the first such number with this property, noting that the question is implicit in Aigner 1961. - Charles R Greathouse IV, Jan 12 2018
Record values begin a(1) = 2, a(7) = 3, a(12) = 4, a(22) = 7, a(30) = 12, a(64) = 25, a(96) = 226, a(330) = 800516; the next record appears to be a(1147), unless a(1147) = -1. (The value for a(330), i.e., for k = 659, is from the Ballinger & Keller link, which also lists k = 2293, i.e., n = (k+1)/2 = (2293+1)/2 = 1147, as the smallest of 50 values of k < 509203 for which no prime of the form k*2^m-1 had yet been found.) - Jon E. Schoenfield, Jan 13 2018
Same as A046069 except for a(2) = 1. - Georg Fischer, Nov 03 2018

Hans Riesel, Några stora primtal, Elementa 39 (1956), pp. 258-260.

Jon E. Schoenfield, Table of n, a(n) for n = 1..329
A. Aigner, Folgen der Art ar^n + b, welche nur teilbare Zahlen liefern, Math. Nachr. 23 (1961), pp. 259-264. (Cited in Browkin & Schinzel)
R. Ballinger & W. Keller, The Riesel Problem: Definition and Status.
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), pp. 55-58.
Wilfrid Keller, List of primes k.2^n - 1 for k < 300 .
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. A040081, A046069.

Array[Function[k, SelectFirst[Range@300, PrimeQ[k 2^# - 1] &]][2 # - 1] &, 102] (* Michael De Vlieger, Jan 12 2018 *)
smk[n_]:=Module[{m=1,k=2n-1},While[!PrimeQ[k 2^m-1],m++];m]; Array[smk,120] (* Harvey P. Dale, Dec 26 2023 *)

forstep(k=1,301,2,n=1;while(!isprime(k*2^n-1),n++);print1(n,","))

Edited by Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 25 2006

Name corrected by T. D. Noe, Feb 13 2011

A187714 Odd numbers m divisible by 3 such that for every k >= 1, m*2^k - 1 has a divisor in the set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331}.

Original entry on oeis.org

7148695169714208807, 17968583418362170239, 26363076126393718191, 57376760867272385247, 67950587841687767283, 73873959473901564111, 81055172741266754727, 96217896533288105991, 104173338506128098489

Offset: 1

Arkadiusz Wesolowski, Mar 17 2011

nonn

Wilfrid Keller (2004, published) gave the first known example.
7148695169714208807 computed in 2017 by the author.
Conjecture: 7148695169714208807 is the smallest Riesel number that is divisible by 3. - Arkadiusz Wesolowski, May 12 2017

Chris Caldwell, The Prime Glossary, Riesel number
Carlos Rivera, Problem 49

Cf. A101036, A187716.

Name changed and entry revised by Arkadiusz Wesolowski, May 11 2017

A213529 Smallest Riesel number that is divisible by the n-th prime.

Original entry on oeis.org

7148695169714208807, 84392786545, 42270067, 1254341, 514389187, 16861093, 1730653, 1730681, 4485343, 790841, 15692699, 992077, 2136283, 1730681, 24683107, 9666029, 9560713, 33282853, 9375479, 14604599, 1247173, 19437853, 34546507, 790841, 3781541, 1715053, 17710319, 45501941

Offset: 2

Arkadiusz Wesolowski, Jun 13 2012

nonn

Some examples of Riesel numbers that are divisible by 3 are in A187714.

For an odd prime p and odd k, if p divides k, then p does not divide k*2^n - 1 for any n.

			1254341 is first Riesel number that is divisible by 11, the 5th prime - so a(5) = 1254341.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary, Riesel number

Cf. A076337, A101036, A187714, A222534.

Offset changed and initial term added by Arkadiusz Wesolowski, May 11 2017

A373801 a(1) = 2; thereafter, if a(n-1) is prime then a(n) = prime(n) + 1; otherwise a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

2, 4, 7, 8, 15, 29, 18, 35, 69, 137, 32, 63, 125, 249, 497, 993, 1985, 3969, 7937, 72, 143, 285, 569, 90, 179, 102, 203, 405, 809, 114, 227, 132, 263, 140, 279, 557, 158, 315, 629, 1257, 2513, 5025, 10049, 20097, 40193, 200, 399, 797, 228, 455, 909, 1817, 3633, 7265, 14529, 29057, 58113, 116225

Offset: 1

N. J. A. Sloane, Aug 05 2024

nonn

Inspired by A374965. Just as the Riesel numbers (A101036 etc.) underlie A374965, so the Sierpinski numbers (A076336 etc.) underlie the present sequence. This means that for both A374965 and the present sequence, it is possible that there are only finitely many prime terms.
What is the next prime after a(1336) = 1486047139543908353?
The next prime in the sequence after a(1336) is the 328-digit prime a(2412) = 11027*2^1075 + 1 =
44637792944394283771459323765390022896709223538983902782431025499369487088325693\
80355294302151494343616855815219642969893790841894306289338825113522293047097809\
14527499539453353195318334412379318970183638103791974206651303944817277532365140\
54865648555402249863235603037071611259242935028448372668756790221309881865220759\
33966337. - Alois P. Heinz, Aug 05 2024
For a(1) any prime, the trajectory converges to this sequence. Just as in A374965, the trajectories appear to converge to a few attractors. In fact it appears that for most values of a(1), the trajectory converges to the present sequence. However, for a(1) = 384 and 767 the trajectories are different. - Chai Wah Wu, Aug 07 2024

N. J. A. Sloane, Table of n, a(n) for n = 1..2000
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]

Cf. A374965, A076336, A101036.

For the primes in this sequence, see A373802 and A373803.

A:=Array(1..1200,0);
t:=2;
A[1]:= t;
for n from 2 to 100 do
if isprime(t) then t:=ithprime(n)+1; else t:=2*t-1; fi;
A[n]:=t;
od:
[seq(A[n],n=1..100)];

Module[{n = 1}, NestList[If[n++; PrimeQ[#], Prime[n] + 1, 2*# - 1] &, 2, 100]] (* Paolo Xausa, Aug 07 2024 *)

from sympy import isprime, nextprime
def A373801_gen(): # generator of terms
    a, p = 2, 3
    while True:
        yield a
        a, p = p+1 if isprime(a) else (a<<1)-1, nextprime(p)
A373801_list = list(islice(A373801_gen(),20)) # Chai Wah Wu, Aug 05 2024

A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

Original entry on oeis.org

3, 5, 3, 549755813881, 7, 5, 3, 17, 47, 13, 11, 41, 7, 5, 3, 97, 31, 29, 2011, 89, 23, 536870869, 19, 17, 79, 13, 11, 73, 7, 5, 3, 193, 191, 61, 59, 953, 439, 53, 179, 433, 47, 173, 43, 41, 167, 37, 163, 929, 31, 29, 67108763, 409, 23, 149, 19, 17, 911, 13, 11, 137

Offset: 1

Labos Elemer, Jul 13 2004

nonn

If 2n-1 is a provable Riesel number (A101036), then there exists a finite set of primes P(2n-1) such that every 2^x-(2n-1) > 0 is divisible by p(x) in P(2n-1). If some 2^x-(2n-1) = p(x), then a(n) = p(x). Otherwise, p(x) is a proper divisor of 2^x-(2n-1), which must be composite, and no a(n) exists.

For example, if n = 254602, then 2n-1 = 509203 is a provable Riesel number. Every 2^x-509203 > 0 is divisible by prime p(x) in P(509203) = {3,5,7,13,17,241}. 2^x-509203 > 0 implies x >= 19 implies 2^x-509203 > 241 >= p(x), so p(x) is a proper divisor and every 2^x-509203 is composite. Hence a(254602) does not exist.

			a(1) = 3 is the first Mersenne prime;
a(64) = 2^47 - 127 = 140737488355201, where 47 = A096502(64), 127 = 2*64 - 1.

T. D. Noe, Table of n, a(n) for n = 1..935

Cf. A096502.

f[n_]:=Module[{lst={},exp=Ceiling[Log[2,1+n]]},While[!PrimeQ[2^exp-n],exp++]; AppendTo[lst,2^exp-n]]; Flatten[f/@Range[1,1001,2]] (* Ivan N. Ianakiev, Mar 08 2016 *)

A182296 Prime Riesel numbers: primes p such that p*2^k - 1 is composite for all positive integers k.

Original entry on oeis.org

509203, 1330207, 2251349, 2554843, 2924861, 3177553, 3292241, 3580901, 3661529, 3661543, 4384979, 6055001, 7576559, 7629217, 8086751, 8643209, 9053711, 9203917, 9545351, 10219379, 10645867, 10913233, 10913681, 11694013, 11942443, 13161283, 14608183, 15627133

Offset: 1

Arkadiusz Wesolowski, Apr 23 2012

nonn

Primes in A101036.

			509203 is the first known prime p for which p*2^k - 1 is composite for all positive integers k, so a(1) = 509203.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..3701
Eric Weisstein's World of Mathematics, Riesel Number

Cf. A101036, A101050.

A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 2, 4, 2, 8, 18, 76, 32, 151, 692, 592, 154, 580, 27365, 11267

Offset: 1

Charles R Greathouse IV, Nov 26 2013

Crocker showed that 2^(2^n) - 1 - 2^a - 2^b is not prime (with n > 2) if a and b are distinct. This sequence demonstrates that the theorem is sharp in the sense that distinctness is required.

If n > 2, then the (largest) prime P(n) = 2^(2^n)-2^a(n)-1 is a de Polignac number (A065381); i.e., P(n)-2^m is not prime. It seems that if n > 6, then |P(n)-2^m| is composite for every natural m and P(n)*2^m-1 is composite (by the dual Riesel conjecture). So if n > 6, then the prime P(n) may be a Riesel number (A182296). For example, the prime P(7) = 2^(2^7)-(2^18+1) is the first candidate (note that 2^18+1 is the smallest de Polignac number of form 2^k+1). Also, by Crocker's theorem, the smallest number of form 2^(2^n)-2^m-1, namely 2^(2^n-1)-1 is a de Polignac number (A006285) and for n > 6 may be a dual Riesel number (A101036). For example, the double Mersenne prime 2^(2^7-1)-1 probably is a dual Riesel number. It is not known whether these are Riesel numbers with a covering set. - Thomas Ordowski, Jan 24 2024

Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.

Cf. A156695.

a(n)=my(N=2^2^n-1);for(a=1,2^n-1,if(ispseudoprime(N-2^a), return(a)));0

a(15) from Charles R Greathouse IV, Dec 02 2013
a(16) from Daniel Suteu, Oct 11 2020
Name edited by Thomas Ordowski, Jan 24 2024

A263561 Odd numbers n such that for every k >= 1, n*2^k - 1 has a divisor in the set {3, 5, 13, 17, 97, 241, 257}.

Original entry on oeis.org

42270067, 97579567, 340716433, 721933559, 890948323, 1726122269, 1865978047, 1889699677, 2362339121, 3185721853, 3637126963, 4668508603, 5064217117, 5569622789, 7480754459, 7701804269, 8594194301, 9005098303, 9180863669, 9939496717, 9979211051

Offset: 1

Arkadiusz Wesolowski, Oct 21 2015

nonn

What is the smallest term of this sequence that belongs to A076335? Is it the smallest Brier number?

This sequence contains only numbers of the form 30*k + 7, 30*k + 11, 30*k + 13, 30*k + 29.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..96
Chris Caldwell, The Prime Glossary, Riesel number
Carlos Rivera, Problem 29 and Problem 58
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A076335, A263347.
Subsequence of A101036.
A263562 gives the primes.

a(n) = a(n-96) + 39832304070 for n > 96.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A187714 Odd numbers m divisible by 3 such that for every k >= 1, m*2^k - 1 has a divisor in the set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331}.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A213529 Smallest Riesel number that is divisible by the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A373801 a(1) = 2; thereafter, if a(n-1) is prime then a(n) = prime(n) + 1; otherwise a(n) = 2*a(n-1) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A182296 Prime Riesel numbers: primes p such that p*2^k - 1 is composite for all positive integers k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.