OFFSET
1,1
COMMENTS
We have the polynomial factorization: n^13+1 = (n+1) * (n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n+1 is prime and n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 is prime.
LINKS
Robert Price, Table of n, a(n) for n = 1..1036
EXAMPLE
2^13+1 = 8193 = 3 * 2731,
22^13+1 = 282810057883082753 = 23 * 12296089473177511,
1030^13+1 = 1468533713451564313811276230000000000001 = 1031 * 1424377995588326201562828545101842871.
MATHEMATICA
Select[Range[0, 300000], PrimeQ[# + 1] && PrimeQ[(#^13 + 1)/(# + 1)] &] (* Robert Price, Mar 11 2015 *)
PROG
(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1600]|IsSemiprime(n^13+1)] // Vincenzo Librandi, Dec 21 2010
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 12 2005
EXTENSIONS
a(19)-a(24) from Vincenzo Librandi, Dec 21 2010
STATUS
approved