Gaps between integers with at most two prime factors: is $\limsup (u_{k+1}-u_k)/\log k=\infty$? (Erdős #1139)
Statement
Let $1\leq u_1<u_2<\cdots$ be the increasing sequence of all positive integers having at most two prime factors counted with multiplicity — that is, $1$, the primes, and the products of exactly two primes ($4,6,9,10,14,15,\ldots$). Is it true that $$\limsup_{k\to\infty}\frac{u_{k+1}-u_k}{\log k}=\infty?$$
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, else full written proof) that $\limsup_{k\to\infty}(u_{k+1}-u_k)/\log k=\infty$, or a disproof establishing a finite upper bound for that limsup. ADVANCES: prove any unconditional statement showing $(u_{k+1}-u_k)/\log k$ is unbounded above along a specified infinite subsequence, or that gaps of size $\gg g(k)$ occur infinitely often for an explicit unbounded $g$; or provide a reproducible computation to a stated record height reporting the largest observed value of $(u_{k+1}-u_k)/\log k$ together with the sieve program (this supports, but does not by itself settle, the limsup claim). Deliver the proof, or the sieve code plus the attained height and the record-gap statistics.
Background
Attributed to Erdős and recorded by Vaughan [Va99, §1.4]; listed as open on erdosproblems.com/1139 (fetched 2026-07-21, status 'open'). The website carries no curated partial results for this problem, so the frontier is essentially the raw question and a solver should re-derive rather than assume prior progress. The sequence of integers with at most two prime factors is OEIS A037143, with A101041 a related sequence; the conjecture asserts that the consecutive gaps $u_{k+1}-u_k$ are, infinitely often, arbitrarily large multiples of $\log k$. Gaps among primes and among squarefree numbers are much-studied SciNet neighbours (for instance erdosproblems.com/208 on squarefree gaps and erdosproblems.com/145 on their moments), but the normalised gap $(u_{k+1}-u_k)/\log k$ for the $E_{\leq 2}$ almost-primes appears to have no known unconditional resolution. No cash prize is attached. Attacker's tool: sieve the sequence to very large heights and track the running maximum of $(u_{k+1}-u_k)/\log k$, looking for divergence — combined with density estimates for numbers with at most two prime factors ($\sim x\log\log x/\log x$) and short-interval sieve bounds.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1139 (T. F. Bloom) | website |
| REF-02 | OEIS A037143 — integers with at most two prime factors | website |
| REF-03 | OEIS A101041 — related sequence for integers with at most two prime factors | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.