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Asymptotic counting of integers with prime factors pra sb

  • Mehdi Golafshan
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Abstract

Let pn be the nth prime, and let r,s≥2 be fixed multiplicatively independent integers. We count the integers up to x whose prime factors all have the form pra sb with integers a,b≥0. Our asymptotic formula for this count has relative error o(1) and is explicit down to the multiplicative constant. For (r,s)=(2,3) these integers are the prime codes of the ordinals below ωω2, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.

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Cite

BibTeXRIS
BibTeX
@misc{golafshan2026asymptotic,
  title = {{Asymptotic counting of integers with prime factors $p_r^a s^b$}},
  author = {Mehdi Golafshan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19434},
  eprint = {2609.19434},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19434},
}
RIS
TY  - UNPB
TI  - Asymptotic counting of integers with prime factors p_r^a s^b
AU  - Mehdi Golafshan
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.19434
UR  - https://arxiv.org/abs/2609.19434
AB  - Let p_n be the nth prime, and let r,s≥2 be fixed multiplicatively independent integers. We count the integers up to x whose prime factors all have the form p_r^a s^b with integers a,b≥0. Our asymptotic formula for this count has relative error o(1) and is explicit down to the multiplicative constant. For (r,s)=(2,3) these integers are the prime codes of the ordinals below ω^ω^2, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.
ER  -