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A105078 Positive integers n such that n^10 + 1 is semiprime. 13

%I

%S 4,16,26,54,110,120,126,260,314,420,444,470,570,646,714,890,946,1010,

%T 1294,1306,1394,1640,1674,1794,1920,1964,2116,2174,2360,2430,2624,

%U 2666,2884,2924,3094,3106,3174,3220,3504,3686,3826,3884,3924,4046,4540,4700

%N Positive integers n such that n^10 + 1 is semiprime.

%C We have the polynomial factorization: n^10+1 = (n^2+1) * (n^8 - n^6 + n^4 - n^2 + 1) Hence after the initial n=1 prime the binomial can only be semiprime if n^2 + 1 is prime and n^8 - n^6 + n^4 - n^2 + 1 is prime.

%H Robert Price, <a href="/A105078/b105078.txt">Table of n, a(n) for n = 1..1105</a>

%e 4^10+1 = 1048577 = 17 * 61681,

%e 16^10+1 = 1099511627777 = 257 * 4278255361,

%e 1010^10+1 = 1104622125411204510010000000001 = 1020101 * 1082855644108970101989901.

%t Select[ Range[5000], PrimeQ[ #^2 + 1] && PrimeQ[(#^10 + 1)/(#^2 + 1)] &] (* _Robert G. Wilson v_, Apr 08 2005 *)

%t Select[Range[4700], PrimeOmega[#^10+1]==2&] (* _Harvey P. Dale_, Jan 13 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 06 2005

%E More terms from _Robert G. Wilson v_, Apr 08 2005

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Last modified April 10 09:20 EDT 2021. Contains 342845 sequences. (Running on oeis4.)