TITLE: When is $2\varphi(n) > n$ – and how to prove it?
QUESTION [6 upvotes]: When coloring the squares of the Ulam spiral not only by black and white (for being prime or non-prime) but by shades of grey representing the normalized totient function $\varphi(n)/n$

and displaying only those numbers with $\varphi(n)/n > 0.5$, i.e. $2\varphi(n) > n$, one finds that for most $n$ – but not all – it holds that 
$$\boxed{2\varphi(n) > n \ \ \equiv\ \  n \text{ is odd }}$$

There are two kinds of possible exceptions:

$2\varphi(n) > n \ \ \text{ and }\ \  n \text{ is even} $
These exceptions would appear as dark crosses, but there are none since $\varphi(n)/n \leq 1/2$ for all even $n$.
$2\varphi(n) \leq n \ \ \text{ and }\ \  n \text{ is odd } $
These exceptions appear as light crosses, and there are some.

The 52 exceptions less than 5000 are 
$105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3003, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 3927, 4095, 4125, 4305, 4389, 4455, 4485, 4515, 4641, 4725, 4785, 4845, 4875, 4935$
They come in two groups:

$n \equiv 0 \pmod{3}$ and $n \equiv 0 \pmod{5}$
$105, 165, 195, 315, 495, 525, 585, 735, 825, 945, 975, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2625, 2805, 2835, 2925, 3045, 3135, 3255, 3315, 3465, 3675, 3705, 3795, 3885, 4095, 4125, 4305, 4455, 4485, 4515, 4725, 4785, 4845, 4875, 4935$
$n \equiv 0 \pmod{3}$ and $n \not\equiv 0 \pmod{5}$ 
$3003, 3927, 4389, 4641$ (yellow squares)

If one could find two characterizations $\phi_1(n)$ and $\phi_2(n)$ for these two sequences, one would have the general result
$$\boxed{2\varphi(n) > n \ \ \equiv\ \  n \text{ is odd } \wedge \neg\phi_1(n) \wedge \neg\phi_2(n)}$$
In the first sequence some kind of regularity is visible:
$$n_{k+1} = n_k + m_k\cdot 30$$
But it's not at all clear to me, how the $m_k$ behave. These are the first values:
$$m_k = 2,1,4,6,1,2,5,3,4,1,6,\dots$$
With the second sequence I'm totally stuck. How can these numbers be characterized?
My question is:

Can someone give characterizations $\phi_1(n)$ and $\phi_2(n)$ such
  that 
$$2\varphi(n) > n \ \ \equiv\ \  n \text{ is odd } \wedge
 \neg\phi_1(n) \wedge \neg\phi_2(n)$$
holds - together with a proof?


The frequency of exceptions seems to be approximately $0.1$. It might be interesting to see, if a better asymptotic limit can be calculated (if there is one).

REPLY [2 votes]: I'd like to sum up some specific parts of Wojowu's answer:

The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$
$\varphi(n)/n < 1/2$ also for

$3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd 

The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$
$\varphi(n)/n < 1/2$ also for

$3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $13 \leq p \leq 23$ prime and $q\in\mathbb{N}^+$ odd
$3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $17 \leq p \leq 19$ prime and $q\in\mathbb{N}^+$ odd



The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

$3\cdot  11 \cdot 13 \cdot 17 \cdot 19  = 138,567$ (is this the next one?)
$3 \cdot 13 \cdot 17 \cdot 19 \cdot  23 \cdot  29 \cdot  31  = 260,468,169$
$5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$