IMO shortlist 1983/2 solution

24th IMO 1983 shortlist

Problem 2

Let s(n) be the sum of the positive divisors of n. For example, s(6) = 1 + 2 + 3 + 6 = 12. Show that there are infinitely many n such that s(n)/n > s(m)/m for all m < n.

Solution

Solution by Demetres Christofides

The sequence s(n)/n is unbounded. For example take n = product of first m primes p₁p₂ ... p_m. Then s(n)/n = (1 + 1/p₁) ... (1 + 1/p_m) > 1 + 1/p₁ + ... + 1/p_m, which diverges. So take the sequence n₁, n₂, ... where n₁ = 1 and n_i is the first i such that s(n_i)/n_i exceeds s(n_i-1)/n_i-1. Then each n_i has the desired property.

24th IMO shortlist 1983