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:contentReference[oaicite:0]index=0. We show that
RDT has asymptotic growth on the order of log log n, independent of prime factorization:contentReference[oaicite:1]index=1:contentReference[oaicite:2]index=2. 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)