A335962 Numbers k such that k^2 + 1 and k^2 + 2 are both squarefree.

1, 2, 3, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 21, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 39, 42, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 60, 61, 62, 64, 65, 66, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 100, 101

Offset: 1

Author

Amiram Eldar, Jul 01 2020

Keywords

• nonn

Comments

Dimitrov (2020) proved that this sequence is infinite and has an asymptotic density Product_{p prime > 2} (1 - ((-1/p) + (-2/p) + 2)/p^2) = 0.67187..., where (a/p) is the Legendre symbol.

Examples

			1 is a term since 1^2 + 1 = 2 and 1^1 + 2 = 3 are both squarefree.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. I. Dimitrov, Pairs of square-free values of the type n^2+1, n^2+2, arXiv:2004.09975 [math.NT], 2020.

Crossrefs

Subsequence of A049533.

Cf. A005117, A007674, A069987.

Programs

Mathematica

Select[Range[100], And @@ SquareFreeQ /@ (#^2 + {1, 2}) &]