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A216270 Numbers n such that n+(n+1), n^2+(n+1)^2, n+(n+1)^2, n^2+(n+1) are all prime.

%I A216270 #26 Apr 21 2017 10:56:40
%S A216270 1,2,5,14,69,99,495,1364,1365,2010,2735,3099,3914,4359,4389,5984,6669,
%T A216270 8435,9164,10794,12075,15224,15315,16014,16470,17900,20214,20769,
%U A216270 21204,23450,24240,26430,26690,27300,29099,35189,38415,38745,42944,47054,48789,50295
%N A216270 Numbers n such that n+(n+1), n^2+(n+1)^2, n+(n+1)^2, n^2+(n+1) are all prime.
%D A216270 Joong Fang, Abstract Algebra, Schaum, 1963, Page 76.
%H A216270 Harvey P. Dale, <a href="/A216270/b216270.txt">Table of n, a(n) for n = 1..1000</a>
%e A216270 n=14:                               29│     │421
%e A216270 n+(n+1)=14+(14+1)=29                   14---196
%e A216270 n^2+(n+1)^2=196+225=421                │  X  │
%e A216270 n+(n+1)^2=14+225=239                   15---225        *15+225+1=241
%e A216270 n^2+(n+1)=196+15=211               211/        \239
%e A216270 .
%e A216270 n=5:                                  11│   │61
%e A216270 n+(n+1)=5+(5+1)=11                      5---25
%e A216270 n^2+(n+1)^2=25+36=61                    │ X │
%e A216270 n+(n+1)^2=5+36=41                       6---36          *6+36+1=43
%e A216270 n^2+(n+1)=25+6=31                    31/      \41
%e A216270 .
%e A216270 n=495:                             991│     │491041
%e A216270 n+(n+1)=495+496=991                   495---245025
%e A216270 n^2+(n+1)^2=491041                    │  X  │
%e A216270 n+(n+1)^2=246511                      496---246016
%e A216270 n^2+(n+1)=245521               245521/       \246511
%e A216270 .
%e A216270 They form the group:
%e A216270 o 1 2 3 (i)
%e A216270 1 0 3 2
%e A216270 2 3 1 0
%e A216270 3 2 0 1
%e A216270 .
%e A216270 For example, for n=99:
%e A216270 99   9801       0 1 2 3 (i)
%e A216270 100  10000
%e A216270 9801  99        1 0 3 2
%e A216270 10000 100
%e A216270 10000 100
%e A216270 99    9801      2 3 1 0
%e A216270 100  10000      3 2 0 1
%e A216270 9801 99
%e A216270 The sum of each column and the sum of each diagonal is a prime number.
%t A216270 Select[Range[51000],AllTrue[{#+(#+1),#^2+(#+1)^2,#+(#+1)^2, #^2+#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Apr 21 2017 *)
%o A216270 (PARI)
%o A216270 is(n) = { isprime(n+(n+1)) & isprime(n^2+(n+1)^2) & isprime(n+(n+1)^2) & isprime(n^2+(n+1)); }
%o A216270 for(n=1,10^6, if (is(n), print1(n,", ")));
%o A216270 /* _Joerg Arndt_, Mar 26 2013 */
%K A216270 nonn
%O A216270 1,2
%A A216270 _César Aguilera_, Mar 15 2013