[Haskell], 64 59 58 57 51 bytes

Saved a total of 6 bytes thanks to @Laikoni and @Ørjan Johansen

(`take`[x|x<-[2..],and[x`gcd`read[c]<2|c<-show x]])

Try it online!

Haskell, 64 59 58 57 51 bytes

Saved a total of 6 bytes thanks to @Laikoni and @Ørjan Johansen

(`take`[x|x<-[2..],and[x`gcd`read[c]<2|c<-show x]])

Try it online!

[Haskell], 64 59 58 57 49 bytes

Saved a total of 8 bytes thanks to @Laikoni and @Ørjan Johansen

`take`[x|x<-[2..],and[x`gcd`read[c]<2|c<-show x]]

Try it online!

[Haskell], 64 59 58 57 51 bytes

Saved a total of 6 bytes thanks to @Laikoni and @Ørjan Johansen

(`take`[x|x<-[2..],and[x`gcd`read[c]<2|c<-show x]])

Try it online!

[Haskell], 64 59 58 57 bytes

flip take[x|x<-[2..],all(<2).map(gcd x.read.pure).show$x]

Try it online!

[Haskell], 64 59 58 57 49 bytes

Saved a total of 8 bytes thanks to @Laikoni and @Ørjan Johansen

`take`[x|x<-[2..],and[x`gcd`read[c]<2|c<-show x]]

Try it online!