A dive into the niche OEIS sequence A000347, which counts partitions of integers into sums of non-integral powers. From A4 = 1 and rapid growth thereafter, the counting can be reformulated as counting ordered quadruples of positive integers x1 ≤ x2 ≤ x3 ≤ x4 with sqrt(x1) + sqrt(x2) + sqrt(x3) + sqrt(x4) ≤ n. We’ll trace the combinatorial setup, explain why the increasing order avoids duplicates, discuss the 2009 Manthar equivalence to the inequality counting problem, and explore the physics connection via Agarwala and Alok’s 1951 paper on statistical mechanics. A concise bridge between pure number theory, combinatorics, and physics.

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