[Sierpiński number](https://en.wikipedia.org/wiki/Sierpinski_number) is an odd integer $k$ such that $2^nk+1$ is composite for all $n\in{\mathbb N}$. In the paper [Sur un probleme concernant les nombres $k\cdot2^n+1$](https://www.e-periodica.ch/digbib/view?pid=edm-001:1960:15#133), zbl:0093.04602 (1960), Sierpiński proved that there are infinitely many $k$ with this property. All integers $2^nk+1$ constructed by Sierpiński are divisible by at least one of the primes in an explicit small set (a [covering set](https://primes.utm.edu/glossary/page.php?sort=SierpinskiNumber)).


Then, in [A note on Sierpinski numbers](https://www.mathstat.dal.ca/FQ/Scanned/33-3/izotov.pdf), zbl:0849.11016 (1995), Izotov has given infinitely many Sierpiński numbers of a new type $(\*)$:


* for $n=4m+2$, the number $2^nk+1$ has an easy algebraic factorization;
* for all other $n$, the number $2^nk+1$ is divisible by at least one of the primes $\{3,17,257,641,65537,6700417\}$ (a covering set).


(The author notes that the above covering set does *not* work when $n=4m+2$.)


Two decades later, in the paper [Sierpiński and Carmichael numbers](https://math.dartmouth.edu/~carlp/BFLPST_AMS121001.pdf), zbl:1325.11010 (2015), 
Banks et al. write:



> 
> **Every** currently known Sierpiński number $k$ possesses at least one covering set $\cal P$,
>  which is a finite set of prime numbers with the property that $2^nk+1$ is divisible
>  by some prime in $\cal P$ for every $n\in{\mathbb N}$. 
> 
> 
> 


So, for Sierpiński numbers of the Izotov type $(\*)$, was a bigger covering set found between 1995 and 2015?



Do Sierpiński numbers of Izotov type have a covering set?

> 
> So, for Sierpiński numbers of the Izotov type $(\*)$, was
> a bigger covering set found between 1995 and 2015?
> 
> 
> 


No, it was not. And it is conjectured that none exists. In the Math Stack Exchange thread linked by Gerry Myerson in a comment to the question, I give other examples of Sierpiński numbers for which it is not likely that they should possess such a covering set $\cal P$.


(I do not know why Banks et al. repeated the obsolete claim that all known Sierpiński numbers possess a covering.)


