cp's OEIS Frontend

In an arbitrarily large pine plantation, a tree with a trunk of radius 1/n is located at each of the lattice points of a square lattice (whose rows are spaced one unit apart), except for one empty lattice point near the center of the plantation. For an observer located at the empty lattice point, how far away is the most distant visible tree trunk? The sequence a(n) is defined as the square of the distance from the observer to the most distant lattice point at which a visible tree trunk is located. (Each tree trunk is assumed to be a vertical cylinder, centered at its respective lattice point. A tree trunk is considered "visible" unless it is completely obscured from view by one or more other tree trunks.)

It is known that, for any coprime x and y, the closest point to the line from (0,0) to (x,y) is 1/sqrt(x^2 + y^2) units away from it (see e.g. the first linked paper in A047896). Since tree trunks intersect lines that are closer than 1/n units, we must have that a(n) < n^2. In addition, a(n) cannot be divisible by the square of any prime p not congruent to 1 modulo 4, since this forces x and y to have common factor p. Combining this with the criteria for a(n) to be a sum of two squares, we have that a(n) is the largest number < n^2 that is either a product of primes congruent to 1 modulo 4 or twice such a product. - Charlie Neder, Jan 15 2019

Does the set of integers included in this sequence eventually include all positive integers other than those that exceed a multiple of 4 by exactly 1?

This concept has been studied under the name "visible lattice points" although the usual version considers the points in an n X n grid that are visible from the origin. - Jeffrey Shallit, May 22 2021