Smallest n-digit prime containing only these digits

Bash + bsd-games package, 28 bytes

• 18 bytes saved thanks to @Dennis.

primes 1|egrep -wm1 [$2]{$1}

Input given at the command-line as n followed by k as a non-delimited list of digits.

Try it online.

Python 2, 66 bytes

f=lambda n,s,k=1,p=1:10**~-n<p%k*k<s>=set(`k`)or-~f(n,s,k+1,p*k*k)

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Takes input like f(3,{'9','3','8'}).

Python has no built-ins for primes, so the function generates them using Wilson's Theorem to check each potential value for k in turn for being prime.

The chained inequality 10**~-n<p%k*k<s>=set(`k`) combines three conditions on k:

• 10**~-n<k: k contains at least n digits. We don't need to check exactly since if we reach more digits, there must have been no solution
• p%k>0: k is prime, via the Wilson's Theorem condition with p=(n-1)!^2. Since p%k is 0 or 1, this can be combined with the previous condition as 10**~-n<p%k*k
• s>=set(`k`): All digits in k are in the set s. This can be spliced in because Python 2 considers sets as bigger than numbers.

If the current k doesn't satisfy all of these, the function recurses onto k+1, adding 1 to the resulting output. Since the output terminates with True which equals 1, and k starts at 1, the output is k. This parallel tracking of k beats outputting k directly on success.

JavaScript (ES7), 100 bytes

Takes input as number of digits n and string of allowed digits s in currying syntax (n)(s). Returns undefined if no solution is found.

Works rather quickly for up to 6 digits, might work for 7 and definitely too slow -- and memory hungry -- beyond that.

n=>s=>(a=[...Array(10**n).keys()]).find(i=>eval(`/[${s}]{${n}}/`).test(i)&&a.every(j=>j<2|j==i|i%j))

Test

let f =

n=>s=>(a=[...Array(10**n).keys()]).find(i=>eval(`/[${s}]{${n}}/`).test(i)&&a.every(j=>j<2|j==i|i%j))
                                        
console.log(f(5)("247")) // 22247