%I #114 Aug 18 2024 22:10:23
%S 509203,762701,777149,790841,992077,1106681,1247173,1254341,1330207,
%T 1330319,1715053,1730653,1730681,1744117,1830187,1976473,2136283,
%U 2251349,2313487,2344211,2554843,2924861,3079469,3177553,3292241,3419789,3423373,3580901
%N Riesel numbers (n*2^k1 is composite for all k>0, n odd) that have a covering set.
%C 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
%C 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
%C 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
%C 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
%C Named after the Swedish mathematician Hans Ivar Riesel (19292014).  _Amiram Eldar_, Jun 20 2021
%C 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
%C 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
%H Pierre CAMI and Arkadiusz Wesolowski, <a href="/A101036/b101036.txt">Table of n, a(n) for n = 1..15000</a> (P. CAMI supplied the first 335 terms)
%H Michael Filaseta, Carrie Finch and Mark Kozek, <a href="https://doi.org/10.1016/j.jnt.2008.02.004">On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture</a>, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 19161940.
%H Michael Filaseta, Jacob Juillerat, and Thomas Luckner, <a href="https://arxiv.org/abs/2209.10646">Consecutive primes which are widely digitally delicate and Brier numbers</a>, arXiv:2209.10646 [math.NT], 2022.
%H Marcos J. González, Alberto Mendoza, Florian Luca, and V. Janitzio Mejía Huguet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Gonzalez/gonz13.html">On Composite Odd Numbers k for Which 2^n * k is a Noncototient for All Positive Integers n</a>, J. Int. Seq., Vol. 24 (2021), Article 21.9.6.
%H Hans Riesel, <a href="/A038699/a038699_1.pdf">Some large prime numbers</a>. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258260) by Lars Blomberg.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riesel_number">Riesel number</a>.
%Y Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.
%Y See A076337 for references and additional information. Cf. A076336.
%K nonn
%O 1,1
%A _David W. Wilson_, Jan 17 2005
%E Up to 3292241, checked by _Don Reble_, Jan 17 2005, who comments that up to this point each n*2^k1 has a prime factor <= 241.
%E New name from _Felix Fröhlich_, Sep 09 2019
