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.
then p>=Prime(k)+ Prime(k+1).
%I A166575 #11 Jun 02 2025 02:09:21
%S A166575 5,13,19,31,37,43,53,61,71,73,79,101,103,113,131,139,157,163,173,191,
%T A166575 193,199,211,223,241,251,269,271,277,293,311,313,331,353,373,379,397,
%U A166575 419,421,439,443,457,463,499,509,521,523,541,577,601,607,613,619,631,653,659,661,673,691
%N A166575 Primes p>=5 with the property: if Prime(k)<p/2<Prime(k+1), then p>=Prime(k)+ Prime(k+1).
%C A166575 If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.
%e A166575 Let p=13. Then we have 5<13/2<7. Since 13>5+7, then 13 is in the sequence.
%t A166575 Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p>=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]
%Y A166575 Cf. A166574, A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554
%K A166575 nonn
%O A166575 1,1
%A A166575 _Vladimir Shevelev_, Oct 17 2009