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A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.

%I A368084 #7 Dec 11 2023 08:37:33
%S A368084 1,13,21,57,73,133,157,273,381,421,553,601,757,813,993,1261,1333,1561,
%T A368084 1641,1893,1981,2257,2353,2653,2757,3081,3193,3541,3661,4033,4161,
%U A368084 5113,5257,5701,5853,6481,6973,7141,7657,7833,8373,8557,9121,9313,9901,10101,10713,10921
%N A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.
%C A368084 Dimitrov (2023) proved that this sequence is infinite.
%H A368084 Amiram Eldar, <a href="/A368084/b368084.txt">Table of n, a(n) for n = 1..10000</a>
%H A368084 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>.
%F A368084 a(n) = A002061(A368083(n) + 1).
%e A368084 1 is a term since 1 is squarefree, 1 = 0^2 + 0 + 1, and 0^2 + 0 + 2 = 2 is also squarefree.
%t A368084 Select[Table[n^2 + n + 1, {n, 0, 100}], And @@ SquareFreeQ /@ {#, #+1} &]
%o A368084 (PARI) lista(kmax) = {my(m); for(k = 0, kmax, m = k^2 + k + 1; if(issquarefree(m) && issquarefree(m + 1), print1(m, ", ")));}
%Y A368084 Intersection of A007674 and A353887.
%Y A368084 Cf. A002061, A005117, A368083.
%K A368084 nonn,easy
%O A368084 1,2
%A A368084 _Amiram Eldar_, Dec 11 2023