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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.
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LINKS
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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
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PROG
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(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
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CROSSREFS
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Cf. A076336, A076335, A003261, A052333, A101036.
Sequence in context: A205167 A252776 A271583 * A258154 A101036 A244070
Adjacent sequences: A076334 A076335 A076336 * A076338 A076339 A076340
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KEYWORD
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nonn,bref,hard,more
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AUTHOR
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N. J. A. Sloane, Nov 07 2002
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EXTENSIONS
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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
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STATUS
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approved
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