We explore A000339, the number A_N of pairs (i1,i2) of positive integers with i1 ≤ i2 and sqrt(i1) + sqrt(i2) ≤ N. This is a non-integral-powers partition problem: we sum square roots, not integers. For each N, A_N counts all such pairs. The sequence begins 1, 5, 18, 45, 100, ... and grows as N increases. The definition and history trace to N. J. A. Sloan (Handbook of Integer Sequences, 1973; OEIS entry, 1995). The topic even connects to physics: Agarwala and Alluk’s 1951 work on statistical mechanics and partitions into non-integral powers. Researchers use Maple and Mathematica to generate many terms and probe asymptotics, illustrating how a quirky counting problem in number theory can link to physics and computation.

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