Semi-prime H-numbers


This problem is based on an exercise of David Hilbert, who pedagogically suggested that one study the theory of $4n+1$ numbers. Here, we do only a bit of that.

An H-number is a positive number which is one more than a multiple of four: $1, 5, 9, 13, 17, 21,...$ are the H-numbers. For this problem we pretend that these are the only numbers. The H-numbers are closed under multiplication.

As with regular integers, we partition the H-numbers into units, H-primes, and H-composites. $1$ is the only unit. An H-number $h$ is H-prime if it is not the unit, and is the product of two H-numbers in only one way: $1 \cdot h$. The rest of the numbers are H-composite.

For examples, the first few H-composites are: $5 \cdot 5 = 25$, $5 \cdot 9 = 45$, $5 \cdot 13 = 65$, $9 \cdot 9 = 81$, $5 \cdot 17 = 85$.

Your task is to count the number of H-semi-primes. An H-semi-prime is an H-number which can be written as the product of exactly two H-primes. The two H-primes may be equal or different. Of the examples above, all five numbers are H-semi-primes. $125 = 5 \cdot 5 \cdot 5$ is not an H-semi-prime, because it is the product of three H-primes.


Each line of input contains an H-number $h \le 1\, 000\, 001$. The last line of input contains $0$ and this line should not be processed. There are at most $10\, 000$ test cases.


For each H-number $h$ in the input, print a line with $h$ followed by the number of H-semi-primes between $1$ and $h$ inclusive, separated by one space in the format shown in the sample.

Sample Input 1 Sample Output 1
21 0
85 5
789 62
CPU Time limit 1 second
Memory limit 1024 MB
Difficulty 5.1medium
Statistics Show
License For educational use only

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