On the ratio of the sum of divisors and Euler’s Totient Function I

Abstract

We prove that the only solutions to the equation σ(n)=2φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n)=2φ(n) with Ω(n)≤ k, and there are at most 22k+k-k squarefree solutions to φ (n)|σ(n) if ω(n)=k. Lastly the number of solutions to φ(n)|φ(n) as x→∞ is O(x exp(-½√log x)).