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.
%I A257933 #30 Jul 20 2019 21:16:36
%S A257933 79,223,439,1087,1223,2399,3023,4759,5927,8647,14159,14639,21023,
%T A257933 24023,25919,28559,31327,33487,42023,47087,56167,61007,64007,67079,
%U A257933 70223,71287,89399,90599,91807,95479,104327,112223,116279,126023,137639,152879,172223,199807
%N A257933 Prime p such that sqrt(p+2) is semiprime (A001358).
%C A257933 The terms are not congruent to 1 (mod 10).
%C A257933 The sequence contains no Mersenne prime p=2^t-1. Since p > 79, t is an odd prime and p+2 = 2^t+1 is divisible by 3. So, since 2^t+1 should be square, 2^t+1 is divisible by 9, i.e., (2^t+1)/3 == 0 (mod 3). (1)
%C A257933 Note that either t=6k+1 or t=6m+5. In each case, (1) is impossible.
%C A257933 Indeed, if t=6k+1, then (2^t+1)/3 = (2*(4^k)^3+1)/3 = (2*(3+1)^(3*k)+1)/3 == (2*binomial(3*k,1)*3+2+1)/3 == 1(mod 3), and analogously in case t=6*m+5, (2^t+1)/3 == 2 (mod 3): a contradiction.
%H A257933 Peter J. C. Moses and Charles R Greathouse IV, <a href="/A257933/b257933.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Moses)
%F A257933 Trivially a(n) >> n^2 log^2 n/(log log n)^2. - _Charles R Greathouse IV_, May 13 2015
%e A257933 Prime 79 is in the sequence because sqrt(79+2) = 9 = 3*3 which is semiprime.
%e A257933 Prime 1223 is in the sequence because sqrt(1223+2) = 35 = 5*7 which is semiprime.
%t A257933 Select[Prime@Range@18000,PrimeOmega[Sqrt[#+2]]==2&]//Quiet (* _Ivan N. Ianakiev_, May 13 2015 *)
%o A257933 (PARI) issemi(n)=bigomega(n)==2
%o A257933 is(n)=isprime(n) && issquare(n+2,&n) && issemi(n) \\ _Charles R Greathouse IV_, May 13 2015
%o A257933 (PARI) list(lim)=my(v=List(), k=sqrt(lim+2), t); forprime(p=2, sqrt(k), forprime(q=p, k\p, if(isprime(t=(p*q)^2-2), listput(v, t)))); Set(v) \\ _Charles R Greathouse IV_, May 13 2015
%o A257933 (Perl) use ntheory ":all"; forprimes { say if is_power($_+2,2) && scalar(factor(sqrtint($_+2)))==2 } 1e7; # _Dana Jacobsen_, May 13 2015
%o A257933 (Perl) use ntheory ":all"; sub list { my($lim,$k,$t,$p,%v)=shift; $k=sqrt($lim+2); forprimes { $p=$_; forprimes { $t=($p*$_)**2-2; $v{$t}++ if is_prime($t); } $p,int($k/$p); } int(sqrt($k)); my @v=sort{$a<=>$b} keys %v; @v; } say for list(1e10); # Translation of PARI, _Dana Jacobsen_, May 13 2015
%Y A257933 Cf. A000040, A001358, A046315.
%K A257933 nonn
%O A257933 1,1
%A A257933 _Vladimir Shevelev_, May 13 2015
%E A257933 More terms from _Peter J. C. Moses_, May 13 2015