OEIS A000298: Number of Partitions into Non-Integral Powers

We explore A000298, the count of ways to sum square roots of positive integers with nondecreasing indices so the total is at most n. For example, a2 = 4 and a3 = 12, with terms like sqrt(1), sqrt(2), ..., and combinations such as sqrt(1)+sqrt(1) or sqrt(1)+sqrt(4) that fit the rule. We'll connect this counting to physics via Agarwala and Alok’s 1951 link to statistical mechanics, trace its appearance in Sloan’s Handbook, and consider what it reveals about using irrational components in partitions—and what might happen with other fractional powers.

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