The definition and key theorems involving Euler's Totient function. For more info, see the "related pages" listed below.
τ(n) is the number of positive divisors of n
Let x = p1e1 p2e2 ... pnen
The factors of x are all the numbers of the form
y = p1d1 p2d2 ... pndn,
where the di each independently range from 0 to ei. There are (1+ei) ways to choose the exponent of the i'th prime factor, so
τ(x) = (1+e1) (1+e2)...(1+en)
τ(k) = n
This is most easily proved by induction on n. It's true when n=1, because τ(1) = 1, and [1/1] = 1.
[(n+1)/h] - [n/h] = 1 or 0, depending on whether h is a divisor of n+1 or not. Therefore,
[(n+1)/h] − n+1
[n/h] = τ(n+1)
And since [n/(n+1)]=0,
[(n+1)/h] − n
[n/h] = τ(n+1),
which establishes the truth of the identity for n+1.
φ(n), the number of coprimes to n: Totient function
μ(n), the M�bius function
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