OEIS A080234: Partitions into non-integral powers

We explore A080234, the counting function for the number of nondecreasing sequences of positive integers (psi1, psi2, ..., psik) with psi1 ≥ 1, k ≤ n, and sum of psi_i^23 ≤ n. Through concrete examples (like a3 = 8) we see how this generalizes ordinary partitions by replacing summands with non-integral-power weights. We’ll trace the historical thread from mid-20th-century physics, where partition-like counts appeared in studies of energy levels and statistical mechanics, to modern work that uses these generalized partitions to probe asymptotics, generating functions, and connections to analytic number theory. This episode highlights how a seemingly niche sequence opens doors to physics, combinatorics, and deep analysis.}

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