Definition:Sierpiński Number of the Second Kind

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 John Selfridge in 1962 that $$78\, 557$$ is Sierpiński, but there are still some numbers smaller than that whose status is uncertain.

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