%I A129829
%S A129829 2,9,12,20,21,25,29,32,48,49,58,62,65,73,75,81,90,101,110,119,122,139,
%T A129829 156,176,181,184,186,189,200
%N A129829 Conjectured first occurrence of numbers n with the property that there 
               exist two consecutive primes p and q such that pq + n is a cube.
%C A129829 It remains to prove that for certain n, pq+n != y^3 for all consecutive 
               primes p and q. This list was computed for p and q with prime indices 
               up to 100000. Also note that this is not a complete list but rather 
               the first occurrence of a soluton. For example, n = 101 has 2 solutions 
               p=29,q=31 for 29*31+101=1000 = 10^3 and p=197,p=199 for 197*199+101 
               = 38304 = 34^3. n square tend to have more solutions.
%e A129829 p=2,q=3,k=2. 2*3+2 = 8 a cube.
%o A129829 (PARI) primecube(n,m) = { local(c,k,x,p1,p2,j); c=0; for(k=1,m, for(x=1,
               n, p1=prime(x); p2=(prime(x+1)); y=p1*p2+k; if(iscube(y), c++; print1(k",
               "); break; ) ) ); c; } iscube(n) = { local(r); r = n^(1/3); if(floor(r+.5)^3== 
               n,1,0) }
%Y A129829 Cf. A129783.
%Y A129829 Sequence in context: A109297 A048768 A070226 this_sequence A053900 A126977 
               A102237
%Y A129829 Adjacent sequences: A129826 A129827 A129828 this_sequence A129830 A129831 
               A129832
%K A129829 easy,nonn
%O A129829 1,1
%A A129829 Cino Hilliard (hillcino368(AT)hotmail.com), May 20 2007