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Trees have many fascinating properties. While this is primarily true for trees in nature, the concept of trees in math and computer science is also interesting. A particular kind of tree, a perfectly balanced tree, is defined as follows.
Every perfectly balanced tree has a positive integer weight. A perfectly balanced tree of weight 1 always consists of a single node. Otherwise, if the weight of a perfectly balanced tree is w and w ≥ 2, then the tree consists of a root node with branches to k subtrees, such that 2 ≤ k ≤ w. In this case, all k subtrees must be completely identical, and be perfectly balanced themselves.
In particular, all k subtrees must have the same weight. This common weight must be the maximum integer value such that the sum of the weights of all k subtrees does not exceed w, the weight of the overall tree. For example, if a perfectly balanced tree of weight 8 has 3 subtrees, then each subtree would have weight 2, since 2 + 2 + 2 = 6 ≤ 8.
Given N, find the number of perfectly balanced trees with weight N.
The input will be a single line containing the integer N (1 ≤ N ≤ 109).
Output a single integer, the number of perfectly balanced trees with weight N.
One tree has a root with four subtrees of weight 1; a second tree has a root with two subtrees of weight 2; the third tree has a root with three subtrees of weight 1.