Number of positive divisiors with gcd condition

[thippli]

Let n be a positive integer and a be a positive divisor of n. Is there any general formula to find the number of positive divisors b of n such that (a,b)=1 ?.

 

[ramsey2879]

No. That would be a formula to factor a number but there isn't any.

 

[Norwegian]

Yes. There is a general formula. First, you should understand the well known formula d(n)=∏(r_i+1) for the number of divisors of n, where n=∏p_i^r_i is the prime factorization of n. See for example wikipedia or OEIS.

In your case, it is the same product as for d(n), except you don't run it over all primes, but exclude the primes dividing a.