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Express n as the sum of k distinct divisors of n

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

August 9, 2011
factorialinductioninequalitiesnumber theory unsolvednumber theory

Problem Statement

(a) For each integer k≥3k\ge 3, find a positive integer nn that can be represented as the sum of exactly kk mutually distinct positive divisors of nn. (b) Suppose that nn can be expressed as the sum of exactly kk mutually distinct positive divisors of nn for some k≥3k\ge 3. Let pp be the smallest prime divisor of nn. Show that 1p+1p+1+⋯+1p+k−1≥1.\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.
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