OFFSET
1,1
COMMENTS
We have the polynomial factorization: n^19 + 1 = (n + 1) * (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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^18 - n^17 + n^16 - n^15 + n^14 - n^13 + 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..1000
FORMULA
a(n)^19 + 1 is semiprime (A001358).
EXAMPLE
2^19 + 1 = 524289 = 3 * 174763,
10^19 + 1 = 10000000000000000001 = 11 * 909090909090909091,
1012^19 + 1 = 125438178100868833265294241234853844232270960601988910249 = 1013 * 1238284087866424810121364671617510801898035149081825373.
MATHEMATICA
Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^19 + 1)/(# + 1)] &] (* Robert Price, Mar 10 2015 *)
Select[Range[5800], PrimeOmega[#^19+1]==2&] (* Harvey P. Dale, Feb 15 2019 *)
PROG
(Magma) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1100]|IsSemiprime(n^19+1)]; // Vincenzo Librandi, Mar 10 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 21 2005
EXTENSIONS
a(12)-a(45) from Robert Price, Mar 09 2015
STATUS
approved