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A104657 Positive integers n such that n^19 + 1 is semiprime (A001358).

%I A104657 #28 Sep 08 2022 08:45:17
%S A104657 2,10,28,106,190,292,556,756,858,906,1012,1030,1032,1060,1372,1450,
%T A104657 1488,1720,1722,1758,1782,1822,1972,2356,2436,2446,2620,2748,2788,
%U A104657 2998,3186,3300,3318,3360,3466,3510,3822,3852,4138,4326,4506,4908,5236,5518,5782
%N A104657 Positive integers n such that n^19 + 1 is semiprime (A001358).
%C A104657 We have the polynomial factorization: n^19 + 1 = (n + 1) * (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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 A104657 Robert Price, <a href="/A104657/b104657.txt">Table of n, a(n) for n = 1..1000</a>
%F A104657 a(n)^19 + 1 is semiprime (A001358).
%e A104657 2^19 + 1 = 524289 = 3 * 174763,
%e A104657 10^19 + 1 = 10000000000000000001 = 11 * 909090909090909091,
%e A104657 1012^19 + 1 = 125438178100868833265294241234853844232270960601988910249 = 1013 * 1238284087866424810121364671617510801898035149081825373.
%t A104657 Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^19 + 1)/(# + 1)] &] (* _Robert Price_, Mar 10 2015 *)
%t A104657 Select[Range[5800],PrimeOmega[#^19+1]==2&] (* _Harvey P. Dale_, Feb 15 2019 *)
%o A104657 (Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1100]|IsSemiprime(n^19+1)]; // _Vincenzo Librandi_, Mar 10 2015
%Y A104657 Cf. A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494.
%K A104657 easy,nonn
%O A104657 1,1
%A A104657 _Jonathan Vos Post_, Apr 21 2005
%E A104657 a(12)-a(45) from _Robert Price_, Mar 09 2015