A105142 Positive integers n such that n^12 + 1 is semiprime.

2, 6, 34, 46, 142, 174, 204, 238, 312, 466, 550, 616, 690, 730, 1136, 1280, 1302, 1330, 1486, 1586, 1610, 1638, 1644, 1652, 1688, 1706, 1772, 1934, 1952, 1972, 2040, 2102, 2108, 2142, 2192, 2238, 2250, 2376, 2400, 2554, 2612, 2646, 3006, 3094, 3550, 3642

Offset

1,1

Comments

We have the polynomial factorization n^12+1 = (n^4+1) * (n^8 - n^4 + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and n^8 - n^4 + 1 is prime.

Links

Robert Price, Table of n, a(n) for n = 1..1515

Formula

a(n)^12+1 is semiprime A001538. a(n)^4+1 is prime and a(n)^8 - a(n)^4 + 1 is prime.

Example

2^12+1 = 4097 = 17 * 241,
6^12+1 = 2176782337 = 1297 * 1678321,
34^12+1 = 2386420683693101057 = 1336337 * 1785792568561,
1136^12+1 = 4618915067251126036363854530631172097 = 1665379926017 * 2773490297975392253706241.

Mathematica

Select[ Range@3691, PrimeQ[ #^4 + 1] && PrimeQ[(#^12 + 1)/(#^4 + 1)] &] (* Robert G. Wilson v *)
Select[Range[4000], PrimeOmega[#^12+1]==2&] (* Harvey P. Dale, Jan 24 2013 *)

Crossrefs

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

Sequence in context: A278611 A088125 A064940 * A227306 A192537 A026966

Adjacent sequences: A105139 A105140 A105141 * A105143 A105144 A105145

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 09 2005

Extensions

a(16)-a(46) from Robert G. Wilson v, Feb 10 2006

Status

approved