A335962 -id:A335962 - cp's OEIS frontend

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

Amiram Eldar, Dec 11 2023

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

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

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.

Subsequence of A353886.

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

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

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

A358428 Numbers k such that k^2 + 1, k^2 + 2 and k^2 + 3 are all squarefree.

Original entry on oeis.org

2, 6, 8, 10, 16, 20, 26, 28, 30, 34, 36, 42, 44, 46, 48, 52, 54, 56, 60, 62, 64, 66, 72, 74, 78, 80, 84, 88, 90, 92, 96, 98, 100, 106, 108, 114, 116, 120, 126, 128, 134, 136, 138, 142, 144, 146, 150, 152, 154, 156, 160, 162, 164, 170, 172, 174, 178, 180, 186, 188, 190, 192, 196, 198, 200

Offset: 1

Michel Marcus, Nov 15 2022

nonn

Wongcharoenbhorn proves that this sequence is infinite and gives an infinite product for its density; its value is about 0.313992945491, so a(n) ~ kn with k around 3.18478492705. - Charles R Greathouse IV, Dec 11 2022

W. Wongcharoenbhorn, Three consecutive near-square squarefree numbers, arXiv:2211.07237 [math.NT], 2022.

Subsequence of A335962.

Select[Range[200], And @@ SquareFreeQ /@ (#^2 + {1, 2, 3}) &] (* Amiram Eldar, Nov 15 2022 *)

isok(k) = issquarefree(k^2+1) && issquarefree(k^2+2) && issquarefree(k^2+3);

cp's OEIS Frontend

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

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