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Abstract: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 (\pi_k) determines the height multiplicities (N_k) through the weighted-multipartition identity (\sum_k N_k q^k=\prod_j(1-q^j)^{-\pi_j}), 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 (\Pi_k\sim Ck^\alpha), then (\log N_k\sim C_2 k^{\alpha/(\alpha+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.

Submission history

From: Hartosh Singh Bal [view email]
[v1] Tue, 1 Aug 2023 11:19:03 UTC (153 KB)
[v2] Tue, 27 Jan 2026 08:06:06 UTC (65 KB)
[v3] Thu, 30 Jul 2026 15:27:52 UTC (63 KB)