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A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).

%I A105282 #14 Sep 08 2022 08:45:17
%S A105282 2,4,46,154,266,472,748,1434,1738,2058,2204,2222,2428,2478,2510,2866,
%T A105282 3132,3288,3576,3688,3756,4142,4506,4940,5164,6252,6330,6786,7180,
%U A105282 7300,7338,7416,7628,7806,9270,9312,10044,10722,10860,12126,12422,12668,12998,13350
%N A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).
%C A105282 We have the polynomial factorization: n^20 + 1 = (n^4 + 1) * (n^16 - n^12 + 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^16 - n^12 + n^8 - n^4 + 1) is prime.
%H A105282 Robert Price, <a href="/A105282/b105282.txt">Table of n, a(n) for n = 1..1405</a>
%F A105282 a(n)^20 + 1 is semiprime (A001358).
%e A105282 2^20 + 1 = 1048577 = 17 * 61681,
%e A105282 4^20 + 1 = 1099511627777 = 257 * 4278255361,
%e A105282 46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,
%e A105282 1434^20 + 1 =
%e A105282 1352019721694375552250489804528860551814233886722212960509362177 =
%e A105282 4228599998737 * 319732233386510278346888399489424537759394853595121.
%t A105282 Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* _Robert Price_, Mar 09 2015 *)
%o A105282 (Magma)IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // _Vincenzo Librandi_, Dec 21 2010
%Y A105282 Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657.
%K A105282 easy,nonn
%O A105282 1,1
%A A105282 _Jonathan Vos Post_, Apr 25 2005
%E A105282 a(9)-a(44) from _Robert Price_, Mar 09 2015