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

1,2

Comments

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

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.

Example

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

Mathematica

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

Prog

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

Crossrefs

Subsequence of A353886.

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

Sequence in context: A058235 A107819 A300950 * A067186 A133675 A332068

Adjacent sequences: A368080 A368081 A368082 * A368084 A368085 A368086

Keyword

nonn,easy

Author

Amiram Eldar, Dec 11 2023

Status

approved