Definition:Sierpiński Number of the Second Kind

Definition
A Sierpiński number is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$.

That is, when $k$ is a Sierpiński number, all members of the set:
 * $\left\{{k 2^n + 1}\right\}$

are composite.

The list of known Sierpiński numbers starts:
 * $78\ 557, \ 271\ 129, \ 271\ 577, \ 322\ 523, \ 327\ 739, \ 482\ 719, \ 575\ 041, \ 603\ 713, \ 903\ 983, \ 934\ 909, \ 965\ 431, \ \ldots$

It has been conjectured that $78\ 557$ is the smallest Sierpiński number.

It was proved by in 1962 that $78\ 557$ is Sierpiński, but there are still some numbers smaller than that whose status is uncertain.

Also see

 * Sierpiński Problem

He proved in 1960 that there is an infinite number of Sierpiński numbers.