OEIS A000345: Partitions into non-integral powers

A journey from a physics-inspired partition problem to a concrete lattice-point counting interpretation. We explore A000345, the nonnegative sequence counting partitions into non-integral powers, with roots in a 1951 statistical mechanics paper by Agarwala and Auluck and entries in Sloan’s Handbooks. In 2009, R.J. Mathar gave a concrete reinterpretation: the partition count equals the number of integer solutions to a radical inequality, turning an abstract partition problem into counting lattice points inside a curved region defined by sums of square roots. We'll connect the early terms 1, 5, 22, 71, 186 to this geometry, and reflect on what this cross-disciplinary link reveals about the unity of number theory, geometry, and physics for students delving into the OEIS.

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