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%I #13 Sep 08 2022 08:45:17
%S 2,22,108,126,180,256,336,490,630,652,660,682,708,760,828,862,882,
%T 1030,1038,1128,1162,1216,1318,1450,1612,1930,1950,2010,2236,2268,
%U 2380,2436,2658,2752,2800,2962,2998,3036,3048,3318,3672,3922,4152,4396,4506,4816
%N Positive integers n such that n^13 + 1 is semiprime.
%C We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.
%H Robert Price, <a href="/A105237/b105237.txt">Table of n, a(n) for n = 1..1036</a>
%e 2^13+1 = 8193 = 3 * 2731,
%e 22^13+1 = 282810057883082753 = 23 * 12296089473177511,
%e 1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.
%t Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* _Robert Price_, Mar 11 2015 *)
%o (Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // _Vincenzo Librandi_, Dec 21 2010
%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 12 2005
%E a(19)-a(24) from _Vincenzo Librandi_, Dec 21 2010
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