

A101036


Riesel numbers (n*2^k1 is composite for all k>0, n odd) that have a covering set.


39



509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, 3079469, 3177553, 3292241, 3419789, 3423373, 3580901
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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
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
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
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
Named after the Swedish mathematician Hans Ivar Riesel (19292014).  Amiram Eldar, Jun 20 2021
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
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


LINKS



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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.


STATUS

approved



