%I #17 Jul 23 2024 02:57:50

%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 Since n^12 + 1 = (n^4+1) * (n^8 - n^4 + 1), n^12 + 1 can be semiprime only if both n^4 + 1 and n^8 - n^4 + 1 are prime.

%H Robert Price, <a href="/A105142/b105142.txt">Table of n, a(n) for n = 1..1515</a>

%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 (semiprimes), 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