2

I've found seven $n$ for which $\tau(n),$ the number 0f divisors of $n,$ coincides with $\varphi(n),$ the "totient" function, number of integers from $1$ to $n$ and coprime to $n.$ Namely $1,3,8,10,18,24,30.$

Are there more? Or a relatively simple proof that this is all of them?

Thanks for any information.

Note: The answer of Wojowu proves the list complete. But it relies on a (to me) deep result for a lower bound on $\varphi(n)$ which holds for $n>1296.$ I'd still be interested in a simpler proof tht the list is complete, if possible.

coffeemath
  • 7,403

1 Answers1

3

This (finite) sequence exists on OEIS as A020488. In the entry a proof is outlined and a link to an article (in Russian) where it was originally proven.

The idea of the proof is to show that $\tau(n)$ is significantly smaller than $\varphi(n)$. There is an easy upper bound $$\tau(n)\leq 2\sqrt{n}.$$ Bounding $\varphi(n)$ from below is more tricky, you can see some arguments for instance here. One bound you can find there is $$\varphi(n) \geq 2\left( \frac{n}{6} \right)^{2/3}.$$ Those two bounds together imply $\tau(n)<\phi(n)$ for $n>1296$.

Wojowu
  • 26,600
  • 1
    $$ \varphi(n) \geq 2 \cdot \left( \frac{n}{6} \right)^{2/3}. $$ Method goes back to Ramanujan, although he did not include this easy application – Will Jagy Feb 13 '20 at 22:52
  • @WillJagy Indeed, that's one of the bounds proven under the linked question. Perhaps I should include what the bound is in my answer. – Wojowu Feb 13 '20 at 23:02