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).