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A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.

%I A387045 #25 Sep 05 2025 04:32:26
%S A387045 5,6,17,18,33,34,35,36,38,50,53,54,63,70,71,72,73,89,90,97,98,102,109,
%T A387045 110,125,126,127,128,129,150,151,165,167,168,178,188,198,209,210,217,
%U A387045 218,219,220,221,222,242,243,257,258,259,260,277,278,285,286,294
%N A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.
%C A387045 Conjecture: This sequence is infinite.
%H A387045 Jianqiang Zhao, <a href="/A387045/b387045.txt">Table of n, a(n) for n = 1..189</a>
%H A387045 Jianqiang Zhao, <a href="http://arxiv.org/abs/2505.06234">The Largest Circle Enclosing n Lattice Points</a>, arXiv:2505.06234 [math.GM], 2025. This is an expanded version of the paper by Jianqiang Zhao, <a href="https://doi.org/10.3390/geometry2030012">The Largest Circle Enclosing n Lattice Points</a>, Geometry, Vol. 2 (2025), 12.
%e A387045 It can be proved that the largest circle enclosing exactly 5 or 6 lattice points in the interior on the xy-plane does not exist. Number 5 is the smallest nonnegative integer having this property and 6 is the next. Therefore, a(1)=5 and a(2)=6.
%e A387045 Here is a brief argument. For details, please see my arxiv paper 2505.06234.
%e A387045 First, let C be the circle going through (-1,0) centered at (1/2,1/2). It passes exactly 8 lattice points and encloses exactly 4. Now with (-1,0) fixed on the circle we can shrink C by an infinitesimal amount to circle C' so that C' only goes through one lattice point (-1,0). Then another infinitesimal perturbation will move C' to include exactly 5 lattice points in its interior. Another infinitesimal perturbation will move C' to include exactly 6 lattice points in its interior. Therefore, if the largest circle enclosing exactly 5 or 6 interior lattice points exists, then its radius is at least sqrt(10)/2.
%e A387045 Second, a geometric argument shows that if the radius of a circle is at least sqrt(10)/2 then it encloses either exactly 4 interior lattice points or at least 7 interior lattice points.
%Y A387045 Cf. A387044, complement of A387045; A192493, A192494, A128006, A128007.
%K A387045 nonn,new
%O A387045 1,1
%A A387045 _Jianqiang Zhao_, Aug 14 2025