Recursive Division Tree: A Log-Log Algorithm for Integer Depth

The Recursive Division Tree (RDT) algorithm is a novel method for measuring the “logarithmic height” of positive integers. We show that RDT has asymptotic growth on the order of log log n, independent of prime factorization. The algorithm provides new characterizations for several classical number-theoretic sequences. In particular, we identify a 95% depth matching property for twin primes, an approximate formula for perfect numbers, a depth bound for Mersenne primes, and other empirical patterns in Goldbach partitions, highly composite numbers, Fibonacci numbers, and prime-depth transition points. Benchmarks confirm RDT(n) ∼ c log log n with c ≈ 2.24 ± 0.22, and the algorithm executes very quickly in practice (about 4 × 10−5 seconds per call).