# [Jelly], 15 [bytes]

<sup>I'm not convinced this is as terse as possible.</sup>

    2‘×$⁸СÆfḟṂṭɗ@/

A monadic Link that accepts a positive integer, \$n\$, and yields a list of the first \$n\$ Sylvester primes.

**[Try it online!][TIO-m0nzhzzh]**

#### How?

    2‘×$⁸СÆfḟṂṭɗ@/ - Link: positive integer, N
    2               - literal two (first value of Current, below)
         С         - collect, repeating...
        ⁸           - ...times: chain's left argument = N
       $            - ...action: last two links as a monad - f(Current):
     ‘              -              increment {Current} -> Current + 1
      ×             -              multiply {that] by {Current} -> Current × (Current + 1)
           Æf       - prime factors (vectorises)
                  / - reduce by:
                 @  -   with swapped arguments:
                ɗ   -     last three links as a dyad - f(right, left):
             ḟ      -       {right} filter discard {left}
              Ṃ     -       minimum of {that}
               ṭ    -       tack to {left}


[Jelly]: https://github.com/DennisMitchell/jelly
[bytes]: https://github.com/DennisMitchell/jelly/wiki/Code-page

[TIO-m0nzhzzh]: https://tio.run/##ASkA1v9qZWxsef//MuKAmMOXJOKBuMOQwqHDhmbhuJ/huYLhua3Jl0Av////Nw "Jelly – Try It Online"