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Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.
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%I #9 Dec 11 2023 08:37:25

%S 0,3,4,7,8,11,12,16,19,20,23,24,27,28,31,35,36,39,40,43,44,47,48,51,

%T 52,55,56,59,60,63,64,71,72,75,76,80,83,84,87,88,91,92,95,96,99,100,

%U 103,104,107,111,112,115,119,120,123,124,127,131,132,135,139,140,143

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

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

%H Amiram Eldar, <a href="/A368083/b368083.txt">Table of n, a(n) for n = 1..10000</a>

%H Stoyan Dimitrov, <a href="https://hal.science/hal-03735444/">Square-free pairs n^2 + n + 1, n^2 + n + 2</a>, HAL preprint, hal-03735444, 2023; <a href="https://www.researchgate.net/publication/362388114_Square-free_pairs_n2n1_n2n2">ResearchGate link</a>.

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

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

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

%Y Subsequence of A353886.

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

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Dec 11 2023