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Numbers k such that k^2 + 1, k^2 + 2 and k^2 + 3 are all squarefree.
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%I #11 Dec 11 2022 19:24:21

%S 2,6,8,10,16,20,26,28,30,34,36,42,44,46,48,52,54,56,60,62,64,66,72,74,

%T 78,80,84,88,90,92,96,98,100,106,108,114,116,120,126,128,134,136,138,

%U 142,144,146,150,152,154,156,160,162,164,170,172,174,178,180,186,188,190,192,196,198,200

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

%C 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

%H W. Wongcharoenbhorn, <a href="https://arxiv.org/abs/2211.07237">Three consecutive near-square squarefree numbers</a>, arXiv:2211.07237 [math.NT], 2022.

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

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

%Y Subsequence of A335962.

%K nonn

%O 1,1

%A _Michel Marcus_, Nov 15 2022