[Sierpiński number][1] 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 $k\cdot2^n+1$][2], 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).

Then, in [A note on Sierpinski numbers][3], 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 later, in the paper [Sierpiński and Carmichael numbers][4], 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?


  [1]: https://en.wikipedia.org/wiki/Sierpinski_number
  [2]: https://www.e-periodica.ch/digbib/view?pid=edm-001:1960:15#133
  [3]: https://www.mathstat.dal.ca/FQ/Scanned/33-3/izotov.pdf
  [4]: https://math.dartmouth.edu/~carlp/BFLPST_AMS121001.pdf