Lebowitz lockard 2025 increasing sequences decreasing prime factors

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N. Lebowitz-Lockard, Increasing sequences with decreasing prime factors, Notes Number Theory Discrete Math. 31 (2025), no. 3, 635--638; DOI 10.7546/nntdm.2025.31.3.635-638. Received 22 April 2025, revised 14 September 2025, accepted and published online 16 September 2025; MSC 11A05, 11A41.

The retained folder-name PDF is the publisher PDF, 4 pages with a text layer, printed pp. 635--638 (physical p. $n$ is printed p. $634+n$). Provenance: retained from the repository's survey download set of September 2026; the download URL was not recorded, but the identifier is the DOI https://doi.org/10.7546/nntdm.2025.31.3.635-638 printed on the first page. 183,395 bytes. The file prints "Copyright © 2025 by the Author. This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/" on its first page, the Creative Commons Attribution 4.0 license.

Read status: claims checked. The abstract and Theorems 1.3 and 1.4 were read clause by clause on the text layer, and the four-line proof of Theorem 1.3 was read; the proof of Theorem 1.4 was not checked.

Contents

For an arithmetic function $f$ let $g_f(x)$ be the largest $k$ for which there is a sequence $a_1<\cdots<a_k\le x$ with $f(a_1)>\cdots>f(a_k)$; the note writes $g^-(x)=g_{P^-}(x)$ for the smallest prime factor $P^-$.

• Pages 635--636 recall Erdős's question for the largest prime factor, with Cambie's bounds as Theorem 1.1 (p. 635), $2\sqrt{x/\log x}\lesssim g(x)\lesssim2\sqrt2\sqrt{x/\log x}$, and Pollack, Pomerance and Treviño's bounds for Euler's function as Theorem 1.2 (p. 636), $x^{0.19}\le g_\varphi(x)\le x\exp(-(\tfrac12+o(1))\sqrt{\log x\log\log x})$; it cites Tao (its [12]) and others for the variants in which $\varphi$ is constant or increasing.
• Theorem 1.3 (p. 636): $g^-(x)\lesssim2\sqrt x/\log x$. Proof (p. 636): every term after the first is composite, so the smallest prime factors are distinct primes at most $\sqrt x$, and $k\le\pi(\sqrt x)+1$.
• Conjecture 1.1 (p. 636): the largest gap between consecutive primes up to $x$ is $\ll(\log x)^2$ (a weak form of Cramér's conjecture, with Granville's papers cited for the discussion).
• Theorem 1.4 (p. 636; proof p. 637): under Conjecture 1.1, $g^-(x)\gg\sqrt x/(\log x)^2$, by products $q_ip_i$ with $p_i$ the first primes above $\sqrt x/2$ and primes $q_i\ge p_i$. Page 637 notes that the Pollack--Pomerance--Treviño sequence gives $g^-(x)\gg x^{0.19}$ unconditionally, and page 638 that an unconditional $g^-(x)=x^{1/2+o(1)}$ would follow from prime gaps of size $x^{o(1)}$.

Compiled scope

The whole four-page note was read on the text layer; only the proof of Theorem 1.3 was followed. Nothing here is independently reviewed.

Bears on. #49, as the 2025 paper the page records as citing Tao without improving the totient maximum: it concerns sequences whose smallest prime factors decrease and quotes the Euler function bounds only as context, so it says nothing about the problem's strictly $\varphi$-increasing sequences.