This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n=1, the smallest k is 3, where (3!/1-1,3!/1+1) = (5,7) = (A001359(2),A006512(2)). For n=3, the smallest k is 6, where (6!/3-1,6!/3+1) = (239,241) = (A001359(17),A006512(17)).
a = {}; Do[k = 1; While[ ! (PrimeQ[(k! - n)/n] && PrimeQ[(k! + n)/n]), k++ ]; AppendTo[a, (k! - n)/n], {n, 1, 6}]; a
a = {}; Do[k = 1; While[ ! (PrimeQ[(k! - n)/n] && PrimeQ[(k! + n)/n]), k++ ]; AppendTo[a, (k! + n)/n], {n, 1, 6}]; a
1!/1 - 1 = 0 is not prime. 2!/1 - 1 = 1 is not prime. 3!/1 - 1 = 5 is prime. Thus, a(1) = 3.
lnk[n_]:=Module[{k=1},While[!PrimeQ[k!/n-1],k++];k]; Array[lnk,80] (* Harvey P. Dale, Aug 31 2015 *)
a(n) = {for(k=1, oo, s=k!/n-1; if(floor(s)==s, if(ispseudoprime(s), return(k)))); }
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