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No. of distinct prime factors and no. of prime factors

Source: China Mathematical Olympiad 2014 Q4

December 22, 2013

inductionlogarithmsalgebranumber theory proposednumber theoryChina

Problem Statement

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