A105066 Positive integers n such that n^8 + 1 is semiprime.

6, 9, 10, 13, 16, 18, 20, 22, 26, 28, 32, 33, 34, 38, 42, 43, 47, 50, 51, 52, 53, 56, 58, 60, 66, 68, 69, 70, 72, 81, 84, 92, 94, 98, 102, 104, 110, 116, 120, 134, 136, 138, 144, 145, 160, 162, 164, 166, 170, 172, 174, 178, 185, 188, 192, 196, 198, 200, 204, 205, 210

Offset: 1

Author

Jonathan Vos Post, Apr 05 2005

Keywords

• easy
• nonn

Comments

n^8 + 1 is an irreducible polynomial over the integers and thus can be prime (1^8+1=2, 2^8+1=257, 4^8+1=65537) as well as semiprime.

Examples

			6^8+1 = 1679617 = 17 * 98801,
16^8+1 = 4294967297 = 641 * 6700417,
72^8+1 = 722204136308737 = 12110113 * 59636449 where the two factors have the same number of digits.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A001538, A103854, A104238, A105041.

Programs

Mathematica

fQ[n_] := Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == {1, 1}; Select[ Range[220], fQ[ #^8 + 1] &] (* Robert G. Wilson v, Apr 06 2005 *)
Select[Range[300],PrimeOmega[#^8+1]==2&] (* Harvey P. Dale, Nov 19 2018 *)

Formula

a(n)^8+1 is an element of A001538.

Extensions

More terms from Robert G. Wilson v, Apr 06 2005