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

1,2

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.

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.

Example

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

Mathematica

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

Crossrefs

Subsequence of A049533.

Cf. A005117, A007674, A069987.

Sequence in context: A138343 A321883 A139371 * A079338 A047405 A175904

Adjacent sequences: A335959 A335960 A335961 * A335963 A335964 A335965

Keyword

nonn

Author

Amiram Eldar, Jul 01 2020

Status

approved