The meaning of the questions: given n, n can be written in the form of at least two consecutive positive integers and the number of species.
Ideas: Let n can be written as a, a +1, a +2 ...... a + k-1's and (a> = 1), i.e., n = (a + a + k-1) * k / 2. If k is odd, then, N = (A + (k-1) / 2) * k If k is even, then (a + a + k-1) is an odd number. So if there are a set of solutions, must correspond to an odd prime. The following proved, a odd prime number other than 1, each odd prime number corresponds to a solution.Let x n is an odd prime number, so that y = n / x. Then the relationship between x and y, only the following two:
(1) y> (x-1) / 2, In this case, n must be able to be written as n = (a + (k-1) / 2) * k of the form. Wherein x = K, y = a - a + (k-1) / 2, this time to meet a = y-(K-1) / 2 = y-(x-1) / 2 is a positive integer;
(2) y <= (x-1) / 2, where n must be able to be written as n = n = (a + a + k-1) * k / 2 of the form. Wherein, y = k / 2, x = a + A + k-1, this time A = (x +1- k) / 2 = (x +1- y * 2) / 2 = (x-1) / 2-y +1> = 1.
Therefore, an odd prime number must correspond to a solution.