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A324176 Integers k such that floor(sqrt(k)) + floor(sqrt(k/3)) divides k.

%I A324176 #16 Jun 19 2021 11:31:54
%S A324176 1,2,6,15,18,24,32,36,45,55,72,78,84,98,105,112,136,144,152,180,198,
%T A324176 220,230,275,336,390,403,462,525,540,608,663,697,756,774,792,836,855,
%U A324176 874,940,980,1050,1092,1144,1166,1265,1368,1392,1500,1525,1586,1638,1755,1782,1848,1904
%N A324176 Integers k such that floor(sqrt(k)) + floor(sqrt(k/3)) divides k.
%C A324176 This sequence is infinite for the same reason that A324175 is: if x-1 > y > 1 satisfies x^2 - 3*y^2 = -2 (x=A001834(j), y=A001835(j+1), j>0), then x < 3*y. Let k = 3*y^2 + m. By the pigeonhole principle there exists a number m belonging to [0, 2*x - 2] such that x + y | 3*y^2 + m, so such a k is a term.
%H A324176 Harvey P. Dale, <a href="/A324176/b324176.txt">Table of n, a(n) for n = 1..1000</a>
%t A324176 Select[Range[2000],Divisible[#,Floor[Sqrt[#]]+Floor[Sqrt[#/3]]]&] (* _Harvey P. Dale_, Jun 19 2021 *)
%o A324176 (PARI) is(n) = n%(floor(sqrt(n)) + floor(sqrt(n/3))) == 0;
%Y A324176 Cf. A001834, A001835, A324175.
%K A324176 nonn
%O A324176 1,2
%A A324176 _Jinyuan Wang_, Mar 08 2019