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This question ( Proving for $n \ge 25$, $p_n > 3.75n$ where $p_n$ is the $n$th prime. ) led me to ask the following .

Take $n>2$ a positive integer . Let $a_1,a_2,\ldots,a_{\phi(n)}$ be all the numbers less than $n$ and coprime with $n$ . Also denote $x=\frac{n}{\phi(n)}$ . Then for which $n$ ,the numbers $a_1,a_2,\ldots,a_{\phi(n)}$ will be separated from each other by the multiples of $x$ : $$0<a_1<x<a_2<2x<\ldots <a_{\phi(n)-1}<x(\phi(n)-1)<a_{\phi(n)}<x\phi(n)=n$$

All I know is that (miraculously) $n=30$ works .

I find this question very intriguing . Thanks for everyone who can help me with this problem .

Community
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    Any prime works too (including $2$). – TonyK Dec 25 '15 at 21:45
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    @ComplexPhi $n=30$ seems to be an interesting number when it is related with the Totient function. When I saw your question I remembered an old question of mine in which 30 was the only counterexample of a conjecture related with the Totient function. Just in case: http://math.stackexchange.com/questions/1258928/eulers-totient-and-divisors-count-function-relationship-when-frac-varphin – iadvd Dec 25 '15 at 23:31
  • It is strange to single out $n=30$ as a (miraculous) working example $-$ according to Chris Dugan's answer, all $n \le 30$ are working examples except $n=21$. – TonyK Dec 27 '15 at 17:09

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Interesting question.

This property is surprisingly common. I was able to rework a script I had written previously and ran through the numbers $n = 2\ldots10000$ and more than 25% of them satisfy this property $(2764 / 10000)$. In the first $1000$, almost half of them satisfy $(465 / 1000)$.

Interestingly, both $1000$ and $10000$ satisfy this property.

The first bunch of values that don't satisfy it are: $21, 33, 35, 39, 42, 51, 55, 57, 63, 65, 66, 69, 70$

Early on it looks to me like this separating property gets broken for composite numbers with relatively large differences between their prime factors, which makes some kind of sense.

I have a text file with these results if you want to see them, although I'm not sure what the best way to share them is.

I have a feeling that there is a group-theoretical reason for these results, but I'll have to think about it some more. I'll let you know if I discover anything else.

Chris Dugan
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