1,1

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 (1929-2014). - 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

Pierre CAMI and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (P. CAMI supplied the first 335 terms)

Michael Filaseta, Carrie Finch and Mark Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.

Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.

Marcos J. González, Alberto Mendoza, Florian Luca, and V. Janitzio Mejía Huguet, On Composite Odd Numbers k for Which 2^n * k is a Noncototient for All Positive Integers n, J. Int. Seq., Vol. 24 (2021), Article 21.9.6.

Wikipedia, Riesel number.

See A076337 for references and additional information. Cf. A076336.

Sequence in context: A271583 A076337 A258154 * A244070 A206430 A182296

Adjacent sequences: A101033 A101034 A101035 * A101037 A101038 A101039

nonn

David W. Wilson, Jan 17 2005

Up to 3292241, checked by Don Reble, Jan 17 2005, who comments that up to this point each n*2^k-1 has a prime factor <= 241.

New name from Felix Fröhlich, Sep 09 2019

approved