Loeschian Spheres

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The Loeschian numbers (OEIS A003136) are numbers of the form , where and are integers. For example, , with (28,2,4) as the related Loeschian triplet.


But first, something else. A different sequence is the Farey sequence, which at order 5 is

In two dimensions, the Farey sequence can make Ford circles, each with the number as a radius, centered above their number, and tangent to the zero line and each other.

Loeschian spheres can be made, each with the number as a curvature (the reciprocal of the radius). With as a Loeschian triplet satisfying , the sphere centers are at one of six locations.


Contributed by: Ed Pegg Jr (September 2015)
Open content licensed under CC BY-NC-SA




[1] On-Line Encyclopedia of Integer Sequences. "A003136." oeis.org/A003136.

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