1466: Unit Fraction Partition

A fraction whose numerator is 1 and whose denominator is a positive integer is called a unit fraction. A representation of a positive rational number p/q as the sum of finitely many unit fractions is called a partition of p/q into unit fractions. For example, 1/2 + 1/6 is a partition of 2/3 into unit fractions. The difference in the order of addition is disregarded. For example, we do not distinguish 1/6 + 1/2 from 1/2 + 1/6. For given four positive integers p, q, a, and n, count the number of partitions of p/q into unit fractions satisfying the following two conditions. The partition is the sum of at most n many unit fractions. The product of the denominators of the unit fractions in the partition is less than or equal to a. For example, if (p,q,a,n) = (2,3,120,3), you should report 4 sinceenumerates all of the valid partitions.

The input is a sequence of at most 200 data sets followed by a terminator. A data set is a line containing four positive integers p, q, a, and n satisfying p,q <= 800, a <= 12000 and n <= 7. The integers are separated by a space. The terminator is composed of just one line which contains four zeros separated by a space. It is not a part of the input data but a mark for the end of the input.

The output should be composed of lines each of which contains a single integer. No other characters should appear in the output. The output integer corresponding to a data set p, q, a, n should be the number of all partitions of p/q into at most n many unit fractions such that the product of the denominators of the unit fractions is less than or equal to a.

2 3 120 3
2 3 300 3
2 3 299 3
2 3 12 3
2 3 12000 7
54 795 12000 7
2 3 300 1
2 1 200 5
2 4 54 2
0 0 0 0

4
7
6
2
42
1
0
9
3

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