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A076337 Riesel numbers: odd numbers n such that for all k >= 1 the numbers n*2^k - 1 are composite. 26
509203 (list; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 aren't 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

REFERENCES

P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.

LINKS

Table of n, a(n) for n=1..1.

R. Ballinger and W. Keller, The Riesel Problem: Definition and Status

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, 16 (2013), #13.9.8.

Tanya Khovanova, Non Recursions

Joe McLean, Brier Numbers

C. Rivera, Brier numbers

Eric Weisstein's World of Mathematics, Riesel numbers

PROG

(MAGMA) P:=[3, 5, 7, 13, 17, 241]; C:=[0, 1, 0, 7, 3, 23]; Ch:=CRT([Modexp(2, C[i], P[i]): i in [1..#C]], P); R:=Ch/2^Valuation(Ch, 2); R; // Arkadiusz Wesolowski, Dec 09 2014

CROSSREFS

Cf. A076336, A076335, A003261, A052333, A101036.

Sequence in context: A205167 A252776 A271583 * A258154 A101036 A244070

Adjacent sequences:  A076334 A076335 A076336 * A076338 A076339 A076340

KEYWORD

nonn,bref,hard,more

AUTHOR

N. J. A. Sloane, Nov 07 2002

EXTENSIONS

Normally I require at least four terms but I am making an exception for this one 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 by M. F. Hasler, Aug 23 2020

STATUS

approved

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Last modified January 25 18:03 EST 2021. Contains 340419 sequences. (Running on oeis4.)