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A259586 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

%I A259586 #15 Aug 10 2024 16:09:34
%S A259586 41,67,70,123,130,205,212,328,350,403,410,444,526,548,555,608,671,700,
%T A259586 724,750,753,806,869,888,898,951,1026,1033,1067,1086,1096,1149,1224,
%U A259586 1231,1265,1291,1294,1347,1376,1429,1489,1504,1545,1571,1574,1627,1709,1716
%N A259586 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.
%C A259586 It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0; the first appearance of 2 is when k = 41.
%H A259586 G. C. Greubel, <a href="/A259586/b259586.txt">Table of n, a(n) for n = 1..10000</a>
%t A259586 z = 12000; r = Sqrt[2]; s = Sqrt[3];
%t A259586 u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
%t A259586 v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
%t A259586 Flatten[Position[u - v, -2]] (* A259584 *)
%t A259586 Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
%t A259586 Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
%t A259586 Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
%t A259586 Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)
%t A259586 Select[Range[2000],Floor[Sqrt[2]Floor[Sqrt[3]#]]-Floor[Sqrt[3]Floor[Sqrt[2]#]]==2&] (* _Harvey P. Dale_, Aug 10 2024 *)
%Y A259586 Cf. A259584, A259585, A259587, A259724, A259725, A259746.
%K A259586 nonn,easy
%O A259586 1,1
%A A259586 _Clark Kimberling_, Jul 15 2015