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 A222563 #17 May 29 2016 14:24:10
%S A222563 5,59,83,239,281,359,443,479,521,599,761,839,1163,1319,1361,1583,1619,
%T A222563 1721,1787,1871,1877,2003,2063,2339,2927,2969,3251,3371,3407,3671,
%U A222563 3767,3917,4001,4013,4229,4283,4397,4451,4463,4649,4679,5147,5261,6287,6329,6659,6689
%N A222563 Primes p such that the sum of divisors (excluding 1 and p - 1) of p - 1 and the sum of divisors (excluding 1 and p + 1) of p + 1 are both prime.
%H A222563 Paolo P. Lava, <a href="/A222563/b222563.txt">Table of n, a(n) for n = 1..1000</a>
%e A222563 83 is in the sequence because: it is prime, the sum of divisors (excluding 1 and 82) of 82 is 2 + 41 = 43, which is prime, and the sum of divisors (excluding 1 and 84) of 84 is 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 139, which is also prime.
%t A222563 Select[Prime[Range[2,900]],AllTrue[{Total[Most[Rest[Divisors[#-1]]]], Total[ Most[Rest[Divisors[#+1]]]]},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, May 29 2016 *)
%o A222563 (PARI) is(n)=isprime(n)&&isprime(sigma(n-1)-n)&&isprime(sigma(n+1)-n-2) \\ _Charles R Greathouse IV_, Feb 25 2013
%Y A222563 Cf. A048050, A085842.
%K A222563 nonn
%O A222563 1,1
%A A222563 _Gerasimov Sergey_, Feb 25 2013
%E A222563 Extended and a(4) and a(6) inserted by _Charles R Greathouse IV_, Feb 25 2013