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

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

Author

Amiram Eldar, Dec 11 2023

Keywords

• nonn
• easy

Comments

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

Examples

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

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

Subsequence of A353886.

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

Programs

Mathematica

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

PARI

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