%N Least number k such that (k^k-n)/(k-n) is prime, or 0 if no such k exists.
%C a(n)=0 is confirmed for k <= 5000.
%e (1^1-2)/(1-2) = 1 is not prime. (3^3-2)/(3-2) = 25 is not prime. (4^4-2)/(4-2) = 254/2 = 127 is prime. Thus a(2) = 4.
%o (PARI) a(n)=for(k=1,5000,if(k!=n,s=(k^k-n)/(k-n);if(floor(s)==s,if(ispseudoprime(s),return(k)))));
%Y Cf. A242787, A242788.
%A _Derek Orr_, May 22 2014
%E We don't normally allow conjectural terms, except in special circumstances. This is one of those exceptions, for if we included only terms that are known for certain, not much of this sequence would remain. - _N. J. A. Sloane_, May 31 2014