猜想σ((n))/n≥1/2

On The Conjecture σ((n))/n≥1/2

本文证明了当k≤7,a1 >a2 > >ak>1,且ai+1 (i=1,2, ,k)是素数时,σ ∏ki=1ai ≥∏ki =1(ai+1 )成立,进而证明了当n素因子个数不超过 7时,猜想σ( (n))

In this paper,we prove σ∏7i=1α_i≥∏7i=1(α_i+1) when α_i+1 (i=1,2,...,7) are primes and α_i>α_i+1≥2 (i=1,2,...,6). Based on that, we also prove the conjecture σ((n))/n≥1/2 when the number of prime divisors of n is no more than 7.