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%I #4 Nov 22 2014 12:34:02
%S 19,29,41,53,67,71,89,97,103,113,127,137,139,151,167,179,181,191,193,
%T 197,199,229,239,241,251,269,281,293,307,311,317,331,349,373,379,383,
%U 389,419,431,461,467,479,487,491,499,503,509,521,523,541
%N Primes p with the property that p^3=n^3+m^3+k^3.
%C 19^3=18^3+10^3+3^3, 29^3=27^3+15^3+11^3, 41^3=33^3+32^3+6^3, ... Prime number power 3 = sum of 3 cubes.
%t q=3;lst={};Do[Do[Do[p=n^q+m^q+k^q;p=p^(1/q);If[PrimeQ[p],AppendTo[lst,p]],{n,m+1,6!}],{m,k+1,3*5!}],{k,5!}];Length[lst];Length[Union[lst]];Take[Union[lst],20]
%t Select[Prime[Range[100]],Length[PowersRepresentations[#^3,3,3]]>1&] (* _Harvey P. Dale_, Nov 22 2014 *)
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Feb 21 2009
%E More terms from _Harvey P. Dale_, Nov 22 2014