Abstract
TextDefine χ(n) recursively by χ(1)=1 and χ(n)=ϕ(n)+χ(n/q) for all integers n>1, where q is the least prime factor of n, and where ϕ is the Euler totient function. We show that χ(n)=ϕ(d)(χ(ℓ)−1)+χ(d), where n=dℓ and the prime factors of d are greater than the prime factors of ℓ. We also show χ(nm)≤χ(n)χ(m) when n and m are coprime numbers. As an application, we show that for all primes p≥11, χ(p2−p)>χ(p2−1). We discuss the interpretation of χ as the clique number of the power graph of a finite cyclic group and the significance of the inequality in this context. VideoFor a video summary of this paper, please visit https://youtu.be/p8finzAEJps.
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