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

%I A368083 #9 Dec 11 2023 08:37:25
%S A368083 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 A368083 52,55,56,59,60,63,64,71,72,75,76,80,83,84,87,88,91,92,95,96,99,100,
%U A368083 103,104,107,111,112,115,119,120,123,124,127,131,132,135,139,140,143
%N A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.
%C A368083 Dimitrov (2023) proved that this sequence is infinite and gave the formula for its asymptotic density.
%H A368083 Amiram Eldar, <a href="/A368083/b368083.txt">Table of n, a(n) for n = 1..10000</a>
%H A368083 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 A368083 0 is a term since 0^2 + 0 + 1 = 1 and 0^2 + 0 + 2 = 2 are both squarefree numbers.
%t A368083 Select[Range[0, 150], And @@ SquareFreeQ /@ (#^2 + # + {1, 2}) &]
%o A368083 (PARI) is(k) = {my(m = k^2 + k + 1); issquarefree(m) && issquarefree(m + 1);}
%Y A368083 Subsequence of A353886.
%Y A368083 Cf. A002061, A005117, A007674, A014206, A335962, A353887, A368084.
%K A368083 nonn,easy
%O A368083 1,2
%A A368083 _Amiram Eldar_, Dec 11 2023