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PreprintCC BY 4.0arXiv

On a problem due to Erdős concerning w1(n)

  • John M. Campbell
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Abstract

Adopting notation from Erdős [Congress. Numer., 1980], we define w1(n) = w(n) for positive integers n so that w(1) = 0 and so that: If n > 1 and n = ∏ pαp, then w(n) = ∑pαp ∥ n n/pαp. By then defining V(x)=|{w(n):1≤ n≤ x}| for x ≥ 1, Erdős considered whether or not V(x) = o(x) and left this as an open problem. This same problem was also given as an open problem in Guy's famous textbook on unsolved problems in number theory and seems to have remained open. We solve this problem by proving that V(x) ≠ o(x), and, moreover, by proving the stronger result such that liminfx → ∞ V(x)/x > 0.

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Cite

BibTeXRIS
BibTeX
@misc{campbell2026problem,
  title = {{On a problem due to Erdős concerning $w_1(n)$}},
  author = {John M. Campbell},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07617},
  eprint = {2611.07617},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07617},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a problem due to Erdős concerning w_1(n)
AU  - John M. Campbell
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07617
UR  - https://arxiv.org/abs/2611.07617
AB  - Adopting notation from Erdős [Congress. Numer., 1980], we define w_1(n) = w(n) for positive integers n so that w(1) = 0 and so that: If n > 1 and n = ∏ p^α_p, then w(n) = ∑_p^α_p ∥ n n/p^α_p. By then defining V(x)=|{w(n):1≤ n≤ x}| for x ≥ 1, Erdős considered whether or not V(x) = o(x) and left this as an open problem. This same problem was also given as an open problem in Guy's famous textbook on unsolved problems in number theory and seems to have remained open. We solve this problem by proving that V(x) ≠ o(x), and, moreover, by proving the stronger result such that liminf_x → ∞ V(x)/x > 0.
ER  -