Let $f$ be defined by $$f(n) = \frac{\phi(n)}{n}.$$ Then define a sequence $(n_k)$ by $$n_1 = 1, \mbox{and for } k \geq 2,$$ $$n_k = \ \mbox{smallest integer }n\ > n_{k-1} \ \mbox{with}\ f(n) > f(n_k)$$ for any $n < n_k$. Deduce a formula for $n_k$ and $f(n_k)$, with proof.
Sol. I try to calaulate some values of the $n_k$ as follows : $$\begin{array}{c|ccccccccccccccccc} n & 1 &2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17 \\ \hline \phi(n) &1&1&2&2&4&2&6&4&6&4&10&4&12&6&8&8&16\\ \hline f(n)&1&\frac{1}{2}&\frac{2}{3}&\frac{2}{4}&\frac{4}{5}&\frac{2}{6}&\frac{6}{7}&\frac{4}{8}&\frac{6}{9}&\frac{4}{10}&\frac{10}{11}&\frac{4}{12}&\frac{12}{13}&\frac{6}{14}&\frac{8}{15}&\frac{8}{16}&\frac{16}{17} \end{array}$$
and so $n_1 = 1, n_2 = 2$,and $n_3 = 6$
To see what $n_k$ should be, it requires computations, and I still do not see potential formula.
Can anyone please suggest the formula, or a more effective way to analyse this problem ?
Erick is working this out with you, good. Eventually, you want two items, PLANAT which is commentary on NICOLAS. The Nicolas paper appeared in the Journal of Number Theory, volume 17 (1983), pages 375-388 and has an English abstract. Here we go, NICOLAS 2012 is an update, better estimates, in English.
In an earlier problem, I proved that Ramanujan's method, the same that gives the Superior Highly Composite numbers and the Colossally Abundant numbers, gives the same sequence that Erick is hinting at in comments. Is the Euler phi function bounded below?