|
%I #11 Dec 11 2023 08:37:02
%S 1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,381,421,
%T 463,553,601,651,703,757,813,871,993,1057,1123,1191,1261,1333,1407,
%U 1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653,2757
%N Squarefree numbers of the form k^2 + k + 1 for some k >= 0.
%C Dimitrov proved that this sequence is infinite.
%C Intersection of A002061 and A005117.
%H Amiram Eldar, <a href="/A353887/b353887.txt">Table of n, a(n) for n = 1..10000</a>
%H Stoyan Ivanov Dimitrov, <a href="https://doi.org/10.1515/gmj-2023-2010">Square-free values of n^2+n+1</a>, Georgian Mathematical Journal, Vol. 30, No. 3 (2023), pp. 333-348; <a href="https://arxiv.org/abs/2205.02488">arXiv preprint</a>, arXiv:2205.02488 [math.NT], 2022-2023.
%F a(n) = A002061(1 + A353886(n)).
%e 4^2 + 4 + 1 = 21 = 3 * 7 is squarefree, so 21 belongs to this sequence.
%t Select[Table[n^2 + n + 1, {n, 0, 52}], SquareFreeQ] (* _Amiram Eldar_, Dec 11 2023 *)
%o (PARI) for (k=0, 52, if (issquarefree(m=k^2+k+1), print1 (m", ")))
%Y Cf. A002061, A005117, A353886 (corresponding k's).
%K nonn,easy
%O 1,2
%A _Rémy Sigrist_, May 09 2022
|
