Problem 413

../

claims/: The 1 claim page of Problem 413, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\omega(m) \leq n$?

Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \omega(m)\leq n$ for all $m<n$?

Status. Open: the site's label (page last edited 17 April 2026). Its commentary credits Lau [La26] with a positive answer to the second question and with a weaker version of the first. The partial claim page Lau 2026 records the result for the second question; the first question has no claim, so the derived standing stays open.

Source. erdosproblems.com/413, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #413, https://www.erdosproblems.com/413.

References.

• [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
• [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.
• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
• [Gu04] Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B8 "Unitary aliquot sequences", p. 98: the Erdős--Selfridge barriers, $n$ with $m+f(m)\le n$ for all $m<n$, the question whether $\omega(m)$ has infinitely many barriers, the list 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 20, 24, 26, 28, 30, ..., and the same question for $\Omega(m)$ with Selfridge's 99840 as the largest barrier below $10^5$. Library home: guy_2004_unsolved_problems_number_theory.
• [La26] C. F. Lau, On the number of prime factors of consecutive integers. arXiv:2604.15042 (v1 16 April 2026, v2 24 June 2026); Theorem 1.3 and Corollary 1.4. Library home: lau_2026_number_prime_factors_consecutive_integers.

Formalization. Statement in formal-conjectures.

Current assessment

Open; one pending partial claim. The site formulation above (page last edited 17 April 2026) asks two questions. The first is whether $\omega$ has infinitely many barriers, Erdős's name in [Er79] for an $n$ with $m+\omega(m)\le n$ for all $m<n$; the second weakens the barrier condition to $m+\epsilon\,\omega(m)\le n$ for some fixed $\epsilon>0$. The parts in the frontmatter are these two questions. Lau's Theorem 1.3 [La26], infinitely many $n$ with $\Omega(n-k)\le C\log k$ for every $1<k<n$, answers the second question yes with $\epsilon=1/(C\log2)$, as the claim page Lau 2026 derives; the preprint has no journal record known here, and the site's commentary crediting it on a problem labeled OPEN is not acceptance, so the claim is pending. The paper's Corollary 1.4, $\omega(n-k)\le k$ for all sufficiently large $k<n$, is the weaker version of the first question that the site's commentary records; it settles no part. The first question is open: the site's commentary reports that Erdős believed $\omega$ and $\Omega$ both have infinitely many barriers, that he proved in [Er79d] that $F(n)=\prod k_i$ for $n=\prod p_i^{k_i}$ has a set of barriers of positive density, that Selfridge found $99840$ to be the largest barrier of $\Omega$ below $10^5$, and that Erdős and Graham [ErGr80] saw the problem as a route to showing that the iteration $n\mapsto n+\omega(n)$ settles into a single sequence from every start, with sieve methods not yet strong enough. Guy's B8 [Gu04] lists the barriers of $\omega$ up to $30$, which are the OEIS sequence A005236 the site cites. The site's thread and proof-claim tab carried no proof claim as of 2026-10-06, and no other result on the problem is known here. Proof coverage: nothing is independently reviewed in this corpus.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1987_locally_repeated_values_certain_arithmetic_functions
• erdos_1987_locally_repeated_values_certain_arithmetic_functions / conjecture_p6
• erdos_1987_locally_repeated_values_certain_arithmetic_functions / theorem_3_2
• erdos_1979_unconventional_problems_number_theory_math_mag
• guy_2004_unsolved_problems_number_theory