The Tusk Series: First Differences of the Sum-of-Prime-Factors Function

We investigate the first-difference sequence of the sum-of-prime-factors-with-multiplicity function sopfr(n) = pa ∥n ap, defining the Tusk Series by T(n) = sopfr(n)−sopfr(n−1) for n≥2. Despite the simplicity of its definition, this sequence exhibits three notable structural features: (1) every prime p produces a large positive spike T(p) = p−sopfr(p−1), giving amplitude-weighted prime detection; (2) exact scale invariance T(kn) = T(n) for all positive integers k, a direct consequence of the complete additivity of sopfr; and (3) intrinsic spectral structure, with dominant Fourier peaks at frequencies 1/pfor small primes, exceeding a 99% AR(1) confidence level by factors of 103. We show that these spectral peaks are harmonically locked to the spectrum of prime gaps—six direct frequency matches and thirty-one harmonic relationships at musical ratios (3:2, 4:3, 2:1) are identified, with significance 6.2σ above a permutation null (p < 0.001). The results are stable across sample sizes from N = 50,000 to N = 500,000. We argue that wave-like structure is intrinsic to the distribution of primes, and describe a physical resonator network tuned to the dominant Tusk frequencies.