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A368083
Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.
2
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
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Dec 11 2023
STATUS
approved