OEIS A000158: Partitions into Non-Integral Powers and a Statistical-Mechanics Connection

We explore A000158, which counts representations of n as sums of terms x^(2/3). We trace how Agarwala and Alok's 1951 work casts these partitions in the language of statistical mechanics, using q-series generating functions to derive asymptotics, and compare Bose–Einstein and Fermi–Dirac statistics. We discuss the first terms 1, 2, 8, 19, 41, and how visualizations hint at normal-like shapes, highlighting surprising bridges between number theory and physics.

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