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%I #11 Mar 11 2015 18:53:37
%S 2,6,34,46,142,174,204,238,312,466,550,616,690,730,1136,1280,1302,
%T 1330,1486,1586,1610,1638,1644,1652,1688,1706,1772,1934,1952,1972,
%U 2040,2102,2108,2142,2192,2238,2250,2376,2400,2554,2612,2646,3006,3094,3550,3642
%N Positive integers n such that n^12 + 1 is semiprime.
%C We have the polynomial factorization n^12+1 = (n^4+1) * (n^8 - n^4 + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and n^8 - n^4 + 1 is prime.
%H Robert Price, <a href="/A105142/b105142.txt">Table of n, a(n) for n = 1..1515</a>
%F a(n)^12+1 is semiprime A001538. a(n)^4+1 is prime and a(n)^8 - a(n)^4 + 1 is prime.
%e 2^12+1 = 4097 = 17 * 241,
%e 6^12+1 = 2176782337 = 1297 * 1678321,
%e 34^12+1 = 2386420683693101057 = 1336337 * 1785792568561,
%e 1136^12+1 = 4618915067251126036363854530631172097 = 1665379926017 * 2773490297975392253706241.
%t Select[ Range@3691, PrimeQ[ #^4 + 1] && PrimeQ[(#^12 + 1)/(#^4 + 1)] &] (* _Robert G. Wilson v_ *)
%t Select[Range[4000],PrimeOmega[#^12+1]==2&] (* _Harvey P. Dale_, Jan 24 2013 *)
%Y Cf. A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Apr 09 2005
%E a(16)-a(46) from _Robert G. Wilson v_, Feb 10 2006
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