Let $\Phi(n) = \varphi(n)/n = \prod_{p|n}(p-1)/p$ be the "normalized totient" of $n$.
Some facts:
$\Phi(p) = (p-1)/p < 1$ for prime numbers with $\lim_{p\rightarrow \infty}\Phi(p) = 1$
$\Phi(n) = 1/2$ iff $n$ is a power of $2$
$\Phi(n) < 1/2$ for all even $n$ that are not powers of $2$ and some odd $n$
if $\Phi(n) > 1/2$ then $n$ is odd
I have some questions concerning numbers with $\Phi(n) < 1/2$:
Are there numbers with arbitrary small $\Phi(n)$? Or is there a lower bound $\Phi_{\text{min}} > 0$?
Are there odd numbers with arbitrary small $\Phi(n)$?
How can this astonishing regularity been explained when displaying in a square spiral only those numbers with $\Phi(n) < 1/3$ – a regular pattern of triples pointing right, down, left, up clockwise (with some irregularily distributed defects of course):
Note that the regular background pattern vanishes when choosing values other than 1/3, e.g. 0.3 (left) or 0.4 (right):
Since the cases $\Phi(n) < 1/2$ and $\Phi(n) < 1/3$ display regular patterns, one might suspect that also $\Phi(n) < 1/5$ gives rise to some regularity. But the numbers envolved in creating that pattern are too big, so I cannot visualize it.
- Supposed one would visualize $\Phi(n) < 1/5$ which regular pattern would emerge (if any)?