Consecutive Positive Integer Sums

Given a positive integer \(N\), determine the number of ways to express \(N\) as a sum of two or more consecutive positive integers.

In other words, you need to find the number of sequences \(a, a+1, a+2, \ldots, a+k\) (with \(k \ge 1\)) such that:

[ N = a + (a+1) + \cdots + (a+k) \quad \text{with} \quad a \ge 1, ]

Note that the sequence must contain at least two integers. For example, when \(N = 15\), there are 3 valid representations:

• \(7 + 8 = 15\)
• \(4 + 5 + 6 = 15\)
• \(1 + 2 + 3 + 4 + 5 = 15\)

Your task is to compute this number for several test cases.

inputFormat

The input is read from standard input and has the following format:

T
N1
N2
... 
NT

Here, \(T\) is the number of test cases, and each \(N_i\) is a positive integer.

outputFormat

For each test case, output one integer on a new line representing the number of ways to express \(N\) as a sum of two or more consecutive positive integers.

## sample

5
15
10
1
5
9

3
1
0
1
2

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