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A103854 Positive integers n such that n^6 + 1 is semiprime.

%I A103854 #19 Aug 31 2021 23:18:51
%S A103854 2,4,10,36,56,94,126,224,260,270,300,350,686,716,780,1036,1070,1080,
%T A103854 1156,1174,1210,1394,1416,1434,1440,1460,1524,1550,1576,1616,1654,
%U A103854 1660,1700,1756,1860,1980,2054,2084,2096,2116,2224,2454,2600,2664,2770,2864
%N A103854 Positive integers n such that n^6 + 1 is semiprime.
%C A103854 n^6+1 can only be prime when n = 1, n^6+1 = 2. This is because the sum of cubes formula gives the polynomial factorization n^6+1 = (n^2+1) * (n^4 - n^2 + 1). Hence n^6+1 can only be semiprime when both (n^2+1) and (n^4 - n^2 + 1) are primes.
%H A103854 Robert Price, <a href="/A103854/b103854.txt">Table of n, a(n) for n = 1..1134</a>
%F A103854 a(n)^6 + 1 is semiprime. (a(n)^2+1) is prime and (a(n)^4 - a(n)^2 + 1) is prime.
%e A103854 n n^6+1 = (n^2+1) * (n^4 - n^2 + 1)
%e A103854 2 65 = 5 * 13
%e A103854 4 4097 = 17 * 241
%e A103854 10 1000001 = 101 * 9901
%e A103854 36 2176782337 = 1297 * 1678321
%e A103854 56 30840979457 = 3137 * 9831361
%e A103854 94 689869781057 = 8837 * 78066061
%e A103854 126 4001504141377 = 15877 * 252031501
%e A103854 224 126324651851777 = 50177 * 2517580801
%t A103854 semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ 2Range@1526, semiprimeQ[ #^6 + 1] &] (* _Robert G. Wilson v_, May 26 2006 *)
%t A103854 Select[Range[200000], PrimeQ[#^2 + 1] && PrimeQ[(#^6 + 1)/(#^2 + 1)] &] (* _Robert Price_, Mar 11 2015 *)
%o A103854 (PARI) is(n)=my(s=n^2); isprime(s+1) && isprime(s^2-s+1) \\ _Charles R Greathouse IV_, Aug 31 2021
%Y A103854 Subsequence of A005574.
%Y A103854 Cf. A001358, A001538, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
%K A103854 easy,nonn
%O A103854 1,1
%A A103854 _Jonathan Vos Post_, Mar 31 2005
%E A103854 More terms from _Robert G. Wilson v_, May 26 2006