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A104657
Positive integers n such that n^19 + 1 is semiprime (A001358).
12
2, 10, 28, 106, 190, 292, 556, 756, 858, 906, 1012, 1030, 1032, 1060, 1372, 1450, 1488, 1720, 1722, 1758, 1782, 1822, 1972, 2356, 2436, 2446, 2620, 2748, 2788, 2998, 3186, 3300, 3318, 3360, 3466, 3510, 3822, 3852, 4138, 4326, 4506, 4908, 5236, 5518, 5782
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
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Apr 21 2005
EXTENSIONS
a(12)-a(45) from Robert Price, Mar 09 2015
STATUS
approved