21 (number)

For any ${\displaystyle a}$ coprime to ${\displaystyle n}$ and ${\displaystyle n-a}$, the condition above holds when one of ${\displaystyle a}$ and ${\displaystyle n-a}$ only has factors ${\displaystyle 2}$ and ${\displaystyle 5}$ (for a representation in base ten).

Let ${\displaystyle A(n)}$ denote the quantity of the numbers smaller than ${\displaystyle n}$ that only have factor ${\displaystyle 2}$ and ${\displaystyle 5}$ and that are coprime to ${\displaystyle n}$, we instantly have ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$.

We can easily see that for sufficiently large ${\displaystyle n}$, ${\displaystyle A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}.}$

However, ${\displaystyle \varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}}$ where ${\displaystyle A(n)=o(\varphi (n))}$ as ${\displaystyle n}$ approaches infinity; thus ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$ fails to hold for sufficiently large ${\displaystyle n}$.

In fact, for every ${\displaystyle n>2}$, we have

${\displaystyle A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}{\text{ }}}$ and

${\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}.}$

So ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$ fails to hold when ${\displaystyle n>273}$ (actually, when ${\displaystyle n>33}$).

Just check a few numbers to see that the complete sequence of numbers having this property is ${\displaystyle \{2,3,4,5,6,7,8,9,11,12,15,21\}.}$