A353886 -id:A353886 - cp's OEIS frontend

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

Rémy Sigrist, May 09 2022

Dimitrov proved that this sequence is infinite.

Intersection of A002061 and A005117.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
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.

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

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

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

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

A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 27, 28, 31, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 71, 72, 75, 76, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 111, 112, 115, 119, 120, 123, 124, 127, 131, 132, 135, 139, 140, 143

Offset: 1

Amiram Eldar, Dec 11 2023

Dimitrov (2023) proved that this sequence is infinite and gave the formula for its asymptotic density.

			0 is a term since 0^2 + 0 + 1 = 1 and 0^2 + 0 + 2 = 2 are both squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Dimitrov, Square-free pairs n^2 + n + 1, n^2 + n + 2, HAL preprint, hal-03735444, 2023; ResearchGate link.

Subsequence of A353886.

Cf. A002061, A005117, A007674, A014206, A335962, A353887, A368084.

Select[Range[0, 150], And @@ SquareFreeQ /@ (#^2 + # + {1, 2}) &]

is(k) = {my(m = k^2 + k + 1); issquarefree(m) && issquarefree(m + 1);}

cp's OEIS Frontend

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

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