A binomial number is a number of the form , where , and are integers. Binomial numbers can be factored algebraically as
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for all ,
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for odd, and
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for all positive integers . For example,
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and
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Rather surprisingly, the number of factors of with and symbolic and a positive integer is given by , where is the number of divisors of and is the divisor function. The first few terms are therefore 1, 2, 2, 3, 2, 4, 2, ... (OEIS A000005).
Similarly, the number of factors of is given by , where is the number of odd divisors of and is the odd divisor function. The first few terms are therefore 1, 1, 2, 1, 2, 2, 2, 1,... (OEIS A001227).
In 1770, Euler proved that if , then every odd factor of
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is of the form . (A number of the form is called a Fermat number.)
If and are primes, then
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