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Relation of no., sum, product of factors of n

Source: 2014 China TST 2 Day 1 Q2

March 20, 2014
functioninductionnumber theoryrelatively primenumber theory proposed

Problem Statement

Given a fixed positive integer a≥9a\geq 9. Prove: There exist finitely many positive integers nn, satisfying: (1)τ(n)=a\tau (n)=a (2)n∣ϕ(n)+σ(n)n|\phi (n)+\sigma (n) Note: For positive integer nn, τ(n)\tau (n) is the number of positive divisors of nn, ϕ(n)\phi (n) is the number of positive integers ≤n\leq n and relatively prime with nn, σ(n)\sigma (n) is the sum of positive divisors of nn.
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