A353886 Nonnegative numbers k such that k^2 + k + 1 is squarefree.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72

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

Cf. A000086, A002061, A005117, A353887 (corresponding squarefree numbers).

Sequence in context: A249830 A300473 A271980 * A191882 A098451 A164710

Adjacent sequences: A353883 A353884 A353885 * A353887 A353888 A353889

Keyword

nonn,easy

Author

Rémy Sigrist, May 09 2022

Status

approved