A104479 Positive integers n such that n^16 + 1 is semiprime (A001358).

3, 4, 9, 12, 14, 16, 18, 20, 26, 29, 40, 41, 48, 58, 70, 73, 81, 87, 92, 96, 104, 111, 113, 114, 118, 122, 130, 140, 142, 144, 146, 150, 157, 162, 164, 167, 168, 172, 173, 184, 187, 192, 194, 195, 199, 200, 202, 208, 220, 230, 232, 244, 253, 256, 266, 278, 292, 295, 298

Offset: 1

Author

Jonathan Vos Post, Apr 18 2005

Keywords

• easy
• nonn

Comments

n^16 + 1 is an irreducible polynomial over Z and thus can be either prime (A006313) or semiprime.

Examples

			3^16 + 1 = 43046722 = 2 * 21523361,
4^16 + 1 = 4294967297 = 641 * 6 700417,
9^16 + 1 = 1853020188851842 = 2 * 926510094425921,
12^16 + 1 = 184884258895036417 = 153953 * 1200913648289,
200^16 + 1 = 6553600000000000000000000000000000001 =
162123499503471553 * 40423504427621041217.
		

Links

• Robert Price, Table of n, a(n) for n = 1..102

Crossrefs

Cf. A006313, A001358, A085722, A096173, A186669, A104238, A103854, A105041, A105066, A105078, A105122, A105142, A105237, A104335, A104479, A104494, A104657, A105282.

Programs

Magma

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..300]|IsSemiprime(n^16+1)] // Vincenzo Librandi, Dec 21 2010

Mathematica

Select[Range[300],PrimeOmega[#^16+1]==2&] (* Harvey P. Dale, Aug 21 2011 *)
Select[Range[1000], 2 == Total[Transpose[FactorInteger[#^16 + 1]][[2]]] &] (* Robert Price, Mar 11 2015 *)

Formula

a(n)^16 + 1 is semiprime (A001358).

Extensions

More terms from Vincenzo Librandi, Dec 21 2010

Corrected (adding 202, 208, and 220) by Harvey P. Dale, Aug 21 2011