A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.

1, 13, 21, 57, 73, 133, 157, 273, 381, 421, 553, 601, 757, 813, 993, 1261, 1333, 1561, 1641, 1893, 1981, 2257, 2353, 2653, 2757, 3081, 3193, 3541, 3661, 4033, 4161, 5113, 5257, 5701, 5853, 6481, 6973, 7141, 7657, 7833, 8373, 8557, 9121, 9313, 9901, 10101, 10713, 10921

Offset: 1

Author

Amiram Eldar, Dec 11 2023

Keywords

• nonn
• easy

Comments

Dimitrov (2023) proved that this sequence is infinite.

Examples

			1 is a term since 1 is squarefree, 1 = 0^2 + 0 + 1, and 0^2 + 0 + 2 = 2 is also squarefree.
		

Links

• 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.

Crossrefs

Intersection of A007674 and A353887.

Cf. A002061, A005117, A368083.

Programs

Mathematica

Select[Table[n^2 + n + 1, {n, 0, 100}], And @@ SquareFreeQ /@ {#, #+1} &]

PARI

lista(kmax) = {my(m); for(k = 0, kmax, m = k^2 + k + 1; if(issquarefree(m) && issquarefree(m + 1), print1(m, ", ")));}

Formula

a(n) = A002061(A368083(n) + 1).