A353887 Squarefree numbers of the form k^2 + k + 1 for some k >= 0.

1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 381, 421, 463, 553, 601, 651, 703, 757, 813, 871, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2551, 2653, 2757

Offset

1,2

Comments

Dimitrov proved that this sequence is infinite.

Intersection of A002061 and A005117.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, Szymon Łukaszyk, and Szymon Tworz, Assembly Theory: Formalizing Assembly Spaces and Discovering Patterns and Bounds, Royal Society Open Science (2026).

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.

Formula

a(n) = A002061(1 + A353886(n)).

Example

4^2 + 4 + 1 = 21 = 3 * 7 is squarefree, so 21 belongs to this sequence.

Maple

select(NumberTheory:-IsSquareFree, [seq(k^2+k+1, k=0..100)]); # Robert Israel, May 27 2026

Mathematica

Select[Table[n^2 + n + 1, {n, 0, 52}], SquareFreeQ] (* Amiram Eldar, Dec 11 2023 *)

Prog

(PARI) for (k=0, 52, if (issquarefree(m=k^2+k+1), print1 (m", ")))

Crossrefs

Cf. A002061, A005117, A353886 (corresponding k's).

Sequence in context: A084537 A002061 A247890 * A063541 A393251 A206246

Adjacent sequences: A353884 A353885 A353886 * A353888 A353889 A353890

Keyword

nonn,easy

Author

Rémy Sigrist, May 09 2022

Status

approved