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Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.
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%I #26 Sep 08 2022 08:46:07

%S 2,3,4,6,7,9,13,18,21,22,58,67,79,90,100,106,111,118,120,144,162,174,

%T 195,204,246,273,279,345,393,403,406,435,436,526,541,567,613,625,636,

%U 702,721,729,736,744,762,763,865,898,961,970,993,1059,1099,1117,1131

%N Numbers n such that (n-1)*n^2-1 and n^2-(n-1) are both prime.

%C Numbers n such that (n^3-n^2-1)*(n^2-n+1) is semiprime.

%C Intersection of A162293 and A055494.

%C Primes in this sequence: 2, 3, 7, 13, 67, 79, 541, 613, 1117, ...

%C Squares in this sequence: 4, 9, 100, 144, 961, ...

%H Giovanni Resta, <a href="/A239115/b239115.txt">Table of n, a(n) for n = 1..10000</a>

%e 13 is in this sequence because (13-1)*13^2-1 = 2027 and 13^2-(13-1) = 157 are both prime.

%t Select[Range[1000], PrimeQ[#^3 - #^2 - 1] && PrimeQ[#^2 - # + 1] &] (* _Giovanni Resta_, Mar 10 2014 *)

%t Select[Range[1200],PrimeOmega[#^5-2#^4+2#^3-2#^2+#-1]==2&] (* _Harvey P. Dale_, Sep 24 2014 *)

%o (PARI) isok(n) = isprime(n^3-n^2-1) && isprime(n^2-n+1); \\ _Michel Marcus_, Mar 10 2014

%o (Magma) k:=1;

%o for n in [1..1000] do

%o if IsPrime(k*(n-1)*n^2-1) and IsPrime(k*n^2-n+1) then

%o n;

%o end if;

%o end for; \\ _Juri-Stepan Gerasimov_, Mar 18 2014

%Y Cf. A162291, A002383, A239135, A239326.

%K nonn

%O 1,1

%A _Ilya Lopatin_ following a suggestion from _Juri-Stepan Gerasimov_, Mar 10 2014,

%E More terms from _Giovanni Resta_, Mar 10 2014