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Distinct exponents in the prime factorization
Abstract
Following Erdős (1982) and Sanna (2019), we study the arithmetic function h(n), which is defined to be the number of distinct exponents in the prime factorization of a positive integer n. Among other things, we show that ∑n≤ xh(ϕ(n)) asymp x((loglog x)/(logloglog x))1/2, where ϕ is the Euler totient function. The key ingredient is the Poisson random model for ω(n,T), the number of the prime divisors of n in a given subset of primes T, which was introduced in a recent work of Ford.
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BibTeX
@misc{gabdullin2026distinct,
title = {{Distinct exponents in the prime factorization}},
author = {Mikhail R. Gabdullin and Vitalii V. Iudelevich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25341},
eprint = {2609.25341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25341},
}
RIS
TY - UNPB
TI - Distinct exponents in the prime factorization
AU - Mikhail R. Gabdullin
AU - Vitalii V. Iudelevich
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25341
UR - https://arxiv.org/abs/2609.25341
AB - Following Erdős (1982) and Sanna (2019), we study the arithmetic function h(n), which is defined to be the number of distinct exponents in the prime factorization of a positive integer n. Among other things, we show that ∑_n≤ xh(ϕ(n)) asymp x((loglog x)/(logloglog x))^1/2, where ϕ is the Euler totient function. The key ingredient is the Poisson random model for ω(n,T), the number of the prime divisors of n in a given subset of primes T, which was introduced in a recent work of Ford.
ER -