OFFSET
1,3
COMMENTS
Dimitrov proved that this sequence is infinite.
The number of terms not exceeding X is Product_{p prime} (1 - A000086(p)/p^2) * X + O(X^(4/5+eps)) (Dimitrov, 2023). The coefficient of X, which is the asymptotic density of this sequence, equals Product_{primes p == 1 (mod 3)} (1 - 2/p^2) = 0.93484201367... . - Amiram Eldar, Dec 11 2023
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Ivanov Dimitrov, Square-free values of n^2+n+1, Georgian Mathematical Journal, Vol. 30, No. 3 (2023), pp. 333-348; arXiv preprint, arXiv:2205.02488 [math.NT], 2022-2023.
EXAMPLE
For k = 4, 4^2 + 4 + 1 = 21 = 3 * 7 is squarefree, so 4 belongs to this sequence.
MATHEMATICA
Select[Range[0, 72], SquareFreeQ[#^2 + # + 1] &] (* Amiram Eldar, Dec 11 2023 *)
PROG
(PARI) is(k) = issquarefree(k^2 + k + 1);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rémy Sigrist, May 09 2022
STATUS
approved