Let τ(n) denote the number of positive divisors of n. Cesàro’s formula says that the number of ordered pairs ≺a,b≻ of integers such that 1 ≤ a,b ≤ n and a,b = n is equal to τ(n2), where a,b is the least common multiple of a and b. We explain this result and related results in terms of a finite commutative semigroup of idempotents.
Pentti Haukkanen (Wed,) studied this question.