A Relation Between Partitions and the Number of Divisors

Since the partitions 1 + 2 + 4 and 7 contain an odd number of summands, they are called odd partitions, whereas the other three partitions are called even. Add the smallest numbers of the odd partitions, 1 + 7= 8, and do the same for the smallest numbers of the even partitions, 1 + 2 + 3 = 6. The difference between these two sums, 8 6 = 2, is exactly the number of divisors of the prime 7. In the sequel, p(n) denotes the sum of the smallest numbers of odd partitions of n minus the smallest numbers of even partitions of n, and d(n) denotes the number of divisors of n. For small numbers n, it is easy to check that p(n) equals d(n). This is not a coincidence; we shall see that it is a general relation between the smallest numbers of partitions into unequal parts and the number of divisors.