%I #8 Jun 07 2015 00:54:54
%S 1,2,1009,3598,4354,9214,11662,15051,15052,15873,15874,19042,21772,
%T 22497,22498,24334,26242,46654,60514,76173,76174,93634,97341,97342,
%U 108898,112893,112894,121101,121102,133954,152098,156813,156814,166462,171393,171394,181473,181474,202498,213441,213442,224674,236193,236194,254013,254014,266253,266254,272482,278781
%N Positive integers m such that none of the four consecutive numbers m, m+1, m+2, m+3 can be written as p^2 + q with p and q both prime.
%C Conjecture: Any term not among 1, 2, 1009, 15051, 15052, 21772 has the form 36*k^2-2 or the form 36*k^2-3, where k is a positive integer.
%C Note that this conjecture implies the conjecture in A258168 since neither 36*k^2-2 nor 36*k^2-3 can be a multiple of 5.
%C For more comments, see the linked message to Number Theory Mailing List.
%H Zhi-Wei Sun, <a href="/A258661/b258661.txt">Table of n, a(n) for n = 1..500</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;ca8e9a05.1506">Re: A new conjecture involving p^2+q</a>, a message to Number Theory Mailing List, May 30, 2015.
%e a(1) = 1 since none of 1,2,3,4 has the form p^2+q with p and q both prime.
%e a(2) = 2 since none of 2,3,4,5 has the form p^2+q with p and q both prime.
%t n=0;Do[Do[If[PrimeQ[m+r-Prime[k]^2],Goto[aa]],{r,0,3},{k,1,PrimePi[Sqrt[m+r]]}];n=n+1;Print[n, " ", m];Label[aa];Continue,{m,1,278781}]
%Y Cf. A000040, A258139, A258140, A258141, A258168.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jun 06 2015