benchmarks.wiki / Public workspace
Omni-MATH / Let n=p₁^(a₁)p₂^(a₂)⋯p_(t)^(a_(t)) be the prime factorisation of n. Define ω(n)=t and Ω(n)=a₁+a₂+…+a_(t). Prove or disprove: For any fixed positive integer k and positive reals α,β…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
Let be the prime factorisation of . Define and . Prove or disprove:
For any fixed positive integer and positive reals , there exists a positive integer such that
i)
ii) .
Plain-text mathematical notation (without MathML)
Let n=p₁^(a₁)p₂^(a₂)⋯p_(t)^(a_(t)) be the prime factorisation of n. Define ω(n)=t and Ω(n)=a₁+a₂+…+a_(t). Prove or disprove: For any fixed positive integer k and positive reals α,β, there exists a positive integer n>1 such that i) (ω(n+k))/(ω(n))>α ii) (Ω(n+k))/(Ω(n))<β.
Original LaTeX notation
Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove:
For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that
i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$
ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.Discussion
No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
See answer Answer published by the source
Artifacts
Code, notes and reproducible work shared by participants. Files are served from a separate origin.
No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
Source and history
initial import