Algebraic Properties and Factorization of a Prime-Detecting Divisor Functional

Recent work by Craig, van Ittersum, and Ono demonstrated that certain quasimodular forms, expressed as combinations of MacMahon partition functions, can exactly detect prime numbers. Inspired by the algebraic structure of the quasimodular Eisenstein space, we introduce and study a canonical arithmetic functional R(n) = (n 2−n+1)σ1(n)−σ3(n), where σk(n) denotes the sum of the k-th powers of the divisors of n. We prove that R(n) ≥ 0 for all integers n > 1, with equality holding if and only if n is prime. Furthermore, we establish exact polynomial factorizations for R(n) evaluated at semiprimes and prime powers, demonstrating that the additive sum of prime factors is deterministically encoded in the multiplicativity defect of this functional. Finally, we utilize Ramanujan’s convolution identity for σ3(n) and Euler’s pentagonal number theorem to express R(n) as a pure recurrence relation, characterizing primality as a structural equilibrium between linear divisor moments. This is Part 1 of the 'Structural Divisor Theory' series