# Question on Euler Totient Function and “Tau”

I was studying here and I stumbled upon 2 problems. On the first one I just can't see the pattern between the numbers, but I've found that for all primes except 3 the equality doesn't work. The second one I don't have a clue about how to start, just giving me this hint should do it.

1) Consider the function $$\Phi$$ the Euler's totient function and let $$\tau: \mathbb{N}\rightarrow\mathbb{N}$$ be $$\tau(n)= \displaystyle\sum_{d|n}1$$. Determine all the integers $$n$$ for which $$\Phi(n) = \tau(n)$$.

2) If $$n \in \mathbb{Z}_{+}^{*}$$, then $$\displaystyle\sum_{k=1}^{2n} \tau(k) - \sum_{k=1}^{n} \lfloor\frac{2n}{k}\rfloor = n$$.

Thanks for the help!

• $$\prod_{p^k \|n} \frac{p^{k-1} (p-1)}{k+1} = ?$$ – reuns Sep 10 at 1:03
• The term "$p^k || n$" is "$p^k$ divides $n$"? Just for notation sake. – user447599 Sep 10 at 1:32
• $\tau(n) = \prod_{p^k \| n} \tau(p^k)$ means it is a multiplicative function – reuns Sep 10 at 1:35

## 1 Answer

Hint: For the second one, notice that $$\lfloor \frac{2n}{k}\rfloor=|\{i\leq 2n:k|i\}|$$ and $$\sum _{k=1}^{2n}\tau (k)=\sum _{i=1}^{2n}|\{k\leq 2n:i|k\}|.$$ Also, notice that the limits in the sum are different, so in between $$n+1$$ and $$2n$$ you are missing a $$1$$[why?].