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.
a={};Do[m=n^5;If[PrimeQ[m-5]&&PrimeQ[m+5],AppendTo[a,n]],{n,0,9!}];a
12 is in the sequence, because 12^3 - 5 = 1723 and 12^3 + 5 = 1733 are primes.
Select[Range[8! ],PrimeQ[ #^3-5]&&PrimeQ[ #^3+5]&]
a={};Do[m=n^7;If[PrimeQ[m-7]&&PrimeQ[m+7],AppendTo[a,n]],{n,0,9!}];a
Select[Range[0,130000],And@@PrimeQ[#^7+{7,-7}]&] (* Harvey P. Dale, Aug 26 2013 *)
36 is in the sequence, because 36^3 - 7 = 46649 and 36^3 + 7 = 46663 are primes.
Select[Range[8! ],PrimeQ[ #^3-7]&&PrimeQ[ #^3+7]&]
Select[Range[15000],AllTrue[#^3+{7,-7},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 28 2020 *)
a={};Do[m=n^11;If[PrimeQ[m-11]&&PrimeQ[m+11],AppendTo[a,n]],{n,0,9!}];a
Select[Range[0,200000],And@@PrimeQ[#^11+{11,-11}]&] (* Harvey P. Dale, Jun 20 2013 *)