Problem 890

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Statement. If $\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\geq 1$,

$$\liminf_{n\to \infty}\sum_{0\leq i<k}\omega_k(n+i)\leq k?$$

Is it true that

$$\limsup_{n\to \infty}\left(\sum_{0\leq i<k}\omega(n+i)\right) \frac{\log\log n}{\log n}=1,$$

where $\omega$ counts the number of distinct prime factors without restriction?

Status. Open.

Source. erdosproblems.com/890, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #890, https://www.erdosproblems.com/890.

References.

• [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428-430.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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_1967_problems_prime_factors_consecutive_integers
• erdos_1967_problems_prime_factors_consecutive_integers / conjecture_p429
• erdos_1967_problems_prime_factors_consecutive_integers / inequality_1