OEIS A000135: Number of Partitions into Non-Integral Powers

In this episode we explore A000135, the number of ways to write a positive integer n as a sum of distinct terms of the form m^(2/3) (the two-thirds power of integers). We unpack what it means to partition with non-integer powers, why the terms must be distinct, and how this unusual rule leads to connections with statistical mechanics—modeling energy levels E_m = m^(2/3)—as discussed in the Agarwal–Alok line of work. We’ll cover why there isn’t a simple closed-form formula for A000135, how recursive descriptions and asymptotic techniques help us understand its growth, and what hints it might offer about deeper links to analytic number theory and primes. Along the way we’ll outline the combinatorial questions that arise and point to open problems and directions for further exploration.

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