OFFSET
1,1
COMMENTS
From Robert Gerbicz: there is no n for which n^k+n+1 is semiprime for k=1,2,3,4,5. Proof: n^5+n+1 = (n^2+n+1)*(n^3-n^2+1), here n^2+n+1 is semiprime, so for n > 1, n^5+n+1 has at least 3 factors, hence not a semiprime.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
a(1) = 84 because 84^4 + 84 + 1 = 49787221 = 11 * 4526111; 84^3 + 84 + 1 = 592789 = 29 * 20441; 84^2 + 84 + 1 = 7141 = 37 * 193; 84^1 + 84 + 1 = 169 = 13^2.
3^4+3+1 = 85 = 5*17 is semiprime, but 3^3+3+1 = 321 is prime, so 3 is not in this sequence.
8^4+8+1 = 4105 = 5 * 821 is semiprime, but 8^3+8+1 = 521 is prime, so 8 is not in this sequence.
20^4+20+1 = 160021 = 17 * 9413 is semiprime, and 20^3+20+1 = 8021 = 13 * 617 is semiprime, but 20^2+20+1 = 421 is prime, so 20 is not in this sequence.
PROG
(PARI) is(n)=vector(4, i, bigomega(n^i+n+1))==[2, 2, 2, 2] \\ Charles R Greathouse IV, Nov 13 2012
(Magma) s:=func<n|&+[d[2]: d in Factorization(n)] eq 2>; [k : k in [2..2500]| forall{i:i in [1, 2, 3, 4]| s(k^i+k+1)}]; // Marius A. Burtea, Feb 11 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Nov 13 2012
STATUS
approved