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Numbers k with the property that the maximum density of points in the integer lattice with all distances at least sqrt(k) is 1/k.
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%I #35 Nov 17 2024 07:29:37

%S 1,2,4,5,8,9,10,16,18,20,32,36,49,81

%N Numbers k with the property that the maximum density of points in the integer lattice with all distances at least sqrt(k) is 1/k.

%C All of these are sums of two squares.

%C These distances can be achieved by points on a tilted square lattice.

%e The smallest density of lattice points with distances at least sqrt(5) has density 1/5, so 5 is in the sequence:

%e o....o.

%e ..o....

%e ....o..

%e .o....o

%e ...o...

%e o....o.

%e But the smallest density of lattice points with distances at least sqrt(13) has density 1/12, so 13 is not in the sequence:

%e o.....o

%e .......

%e ...o...

%e .......

%e o.....o

%e .......

%e ...o...

%Y Subsequence of A001481.

%K nonn,fini,full

%O 1,2

%A _Erich Friedman_, Nov 04 2024