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This question is a follow up to this excellent mathematics stackexchange question.

Let $\mu(n)$ be the Möbius function, $\phi(n)$ Euler's totient function, $\sigma(n)$ the sum of divisors function and $\tau(n)$ the number of divisors function. Define the set $S_N,$ for a natural number $N,$ by

$$S_N = \lbrace (m,n) \in \mathbb{N} \times \mathbb{N} \mid m \ne n, \, \mu(m)=\mu(n), \, \phi(m)=\phi(n),$$ $$\sigma(m)=\sigma(n), \, \tau(m)=\tau(n) \textrm{ and } \text{max} \lbrace m,n \rbrace \le N \rbrace .$$

How large is the set $ S_N $ ?

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    "Order" is a really overloaded word; what's wrong with "size"?2010-11-22
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    @Qiaochu Yes, there's the tacit assumption that there are an infinity of pairs satisfying the given relationships, but at the moment I'll hedge my bets in that direction. Although, of course, any information on the size of $S_N$ is very welcome.2010-11-23
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    Have you found any such $m,n$?2011-07-06
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    Are there any squarefree examples (that is, $\mu(m)=\mu(n)\ne0$)?2012-01-12

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