OEIS A000148: Partitions into Non-Integral Powers

We dive into OEIS A000148, exploring partitions into non-integral powers of integers. We break down what non-integral powers mean, using examples like inequalities involving fractional exponents (for instance, x1^{2/3} + x2^{2/3} ≤ n with 1 ≤ x1 ≤ x2) and explain how A000148 counts such solutions. We look at the first terms (1, 2, 7, 15, 28, 45, 70), discuss why a neat closed form is elusive, and examine tools like generating functions and asymptotics that help describe the sequence's growth. The episode also covers computational approaches (including Mathematica code) for generating more terms and highlights exciting connections to physics through Agarwala and Alluk’s work on statistical mechanics and partitions. Finally, we touch on potential applications of fractional exponents in probability, finance, and beyond, and reflect on how these partitions illuminate deep bridges between number theory and physics.

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