2023/08/01 by Hartosh Singh Bal, Bal, Hartosh Singh
Mathematics · #(Primary) 11A25 #(Secondary) 11A41 #05A17 #11N37 #11N56 #11N60 #11P81 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.CO #math.NT #msc:05A17 #msc:11A25 #msc:11A41 #msc:11N37 #msc:11N56 #msc:11N60 #msc:11P81
paper · pdf · doi:10.48550/arxiv.2308.00455
Accepted version. To appear in Integers
openalex publication_date 2023/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/07/30 · arxiv updated 2026/07/31
The height (H(n)) of an integer (n) is classically the number of iterations of Euler's totient function required to reach (1). H. N. Shapiro showed that a modification of this function is completely additive. We study completely additive height functions with finite prime fibers. Their prime-height profile (πk) determines the height multiplicities (Nk) through the weighted-multipartition identity (∑k Nk qk=∏j(1-qj)-πj), and conversely every profile containing infinitely many primes is realizable. We introduce iteratively defined heights encompassing Shapiro-type totient heights and the Matula height. For the Matula height, we give purely number-theoretic proofs of the classical Gutman-Ivi'c extremal bounds, thereby answering their question whether the maximal bound can be derived without recourse to the rooted-tree interpretation. Using Meinardus' theorem in its full form, we prove a conditional inverse-growth law: if (Πk∼ Ckα), then (log Nk∼ C2 kα/(α+1)), with an explicit constant. We also derive average-order results for a canonical sequential realization and report computations for the Shapiro height beyond the polynomial regime.