OPEN
This is open, and cannot be resolved with a finite computation.
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 $m<n$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
In
[Er79] Erdős calls such an $n$ a 'barrier' for $\omega$. Some other natural number theoretic functions (such as $\phi$ and $\sigma$) have no barriers because they increase too rapidly. Erdős believed that $\omega$ should have infinitely many barriers. In
[Er79d] he proves that $F(n)=\prod k_i$, where $n=\prod p_i^{k_i}$, has infinitely many barriers (in fact the set of barriers has positive density in the integers).
Erdős also believed that $\Omega$, the count of the number of prime factors with multiplicity), should have infinitely many barriers. Selfridge found the largest barrier for $\Omega$ which is $<10^5$ is $99840$.
In
[ErGr80] this problem is suggested as a way of showing that the iterated behaviour of $n\mapsto n+\omega(n)$ eventually settles into a single sequence, regardless of the starting value of $n$ (see also
[412] and
[414]).
Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'.
See also
[647] and
[679].
The sequence of barriers for $\omega$ is
A005236 in the OEIS.
This is discussed in problem B8 of Guy's collection
[Gu04].
View the LaTeX source
This page was last edited 07 April 2026. View history
External data from
the database - you can help update this
Formalised statement?
Yes
Related OEIS sequences:
A005236
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Likes this problem
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conglu,
RaziqStark
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Interested in collaborating
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Currently working on this problem
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conglu,
RaziqStark
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This problem looks difficult
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None
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This problem looks tractable
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RaziqStark
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The results on this problem could be formalisable
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I am working on formalising the results on this problem
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #413, https://www.erdosproblems.com/413, accessed 2026-04-11
