Definition:Sierpiński Number of the Second Kind/Sequence

Definition
The sequence of known Sierpiński numbers of the second kind 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 of the second kind.

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