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