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A105282 Positive integers n such that n^20 + 1 is semiprime (A001358). 11
2, 4, 46, 154, 266, 472, 748, 1434, 1738, 2058, 2204, 2222, 2428, 2478, 2510, 2866, 3132, 3288, 3576, 3688, 3756, 4142, 4506, 4940, 5164, 6252, 6330, 6786, 7180, 7300, 7338, 7416, 7628, 7806, 9270, 9312, 10044, 10722, 10860, 12126, 12422, 12668, 12998, 13350 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

We have the polynomial factorization: n^20 + 1 = (n^4 + 1) * (n^16 - n^12 + 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^16 - n^12 + n^8 - n^4 + 1) is prime.

LINKS

Robert Price, Table of n, a(n) for n = 1..1405

FORMULA

a(n)^20 + 1 is semiprime (A001358).

EXAMPLE

2^20 + 1 = 1048577 = 17 * 61681,

4^20 + 1 = 1099511627777 = 257 * 4278255361,

46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,

1434^20 + 1 =

1352019721694375552250489804528860551814233886722212960509362177 =

4228599998737 * 319732233386510278346888399489424537759394853595121.

MATHEMATICA

Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)

PROG

(MAGMA)IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // Vincenzo Librandi, Dec 21 2010

CROSSREFS

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657.

Sequence in context: A007596 A050588 A217795 * A018325 A099804 A019596

Adjacent sequences:  A105279 A105280 A105281 * A105283 A105284 A105285

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Apr 25 2005

EXTENSIONS

a(9)-a(44) from Robert Price, Mar 09 2015

STATUS

approved

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Last modified September 19 15:43 EDT 2020. Contains 337178 sequences. (Running on oeis4.)