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A239135 Numbers k such that (k-1)*k^2 + 1 and k^2 + (k-1) are both prime.

%I A239135 #38 Sep 08 2022 08:46:07
%S A239135 2,3,5,6,8,13,21,24,26,28,35,45,48,50,55,76,83,89,93,96,100,101,115,
%T A239135 120,138,140,148,149,181,191,195,203,206,209,215,230,258,259,281,285,
%U A239135 294,301,309,323,330,349,358,373,380,386,393,395,423,428,433,474,495
%N A239135 Numbers k such that (k-1)*k^2 + 1 and k^2 + (k-1) are both prime.
%C A239135 Numbers k such that (k^3 - k^2 + 1)*(k^2 + k - 1) is semiprime.
%C A239135 Intersection of A045546 and A111501.
%C A239135 Primes in this sequence: 2, 3, 5, 13, 83, 89, 101, 149, 181, 191, ...
%H A239135 Daniel Starodubtsev, <a href="/A239135/b239135.txt">Table of n, a(n) for n = 1..10000</a>
%e A239135 2 is in this sequence because (2-1)*2^2+1=5 and 2^2+(2-1)=5 are both prime.
%t A239135 Select[Range[600],PrimeQ[#^2+#-1]&&PrimeQ[#^2(#-1)+1]&] (* _Farideh Firoozbakht_, Mar 17 2014 *)
%o A239135 (Magma) k := 1;
%o A239135      for n in [1..10000] do
%o A239135         if IsPrime(k*(n - 1)*n^2 + 1) and IsPrime(k*n^2 + n - 1) then
%o A239135            n;
%o A239135         end if;
%o A239135      end for;
%Y A239135 Cf. A239115.
%K A239135 nonn
%O A239135 1,1
%A A239135 _Ilya Lopatin_ following a suggestion from _Juri-Stepan Gerasimov_, Mar 15 2014