Number of ordered pairs of coprime integers from $1$ to $N$

Question

How many ordered pairs $(a, b)$ of positive integers $a$ and $b$ are there such that $\gcd(a, b)= 1$ and $1\leq a,b\leq N$?

My approach is as follows.

Note that $a= b$ only when $a= b= 1$. So this will always give one solution. Now consider the case when $a$ is not equal to $b$. By symmetry assume $a>b$. We can multiply by $2$ later go get our final answer. Now for $N$ let the answer be $f(N)$. Now we need to find the number of ordered pairs $(a, b)$ of positive integers $a$ and $b$ are their such that $\gcd(a, b)= 1$ and $1\leq a<b\leq N+1$. Note that $f(N+1)= f(N) + \phi(N+1)$. This is because $1\leq a<b\leq N$ has $f(N)$ solutions, so $1\leq a<b\leq N+1$ also has these $N$ solutions, but in addition when calculating $f(N+1)$ we can set $b= N+1$ and get $\phi(N)$ possible values of $N$. So $f(N+1)= f(N) + \phi(N)$. Thus solving this recurrence relation we get $$f(N)= \phi(N) + \phi(N-1) + \dots + \phi(4) + f(3)= \phi(N) + \phi(N-1) + \dots + \phi(4) + 2.$$

My question is, is my approach correct here? Can we further simplify $$\phi(N) + \phi(N-1) + \dots + \phi(4) + f(3)= \phi(N) + \phi(N-1) + \dots + \phi(4) + 2\text{ ?}$$

Note:- the original question was to find the number of ordered pairs $(a, b)$ of positive integers $a,b$ such that $1\leq a,b\leq 50$ and $\gcd(a, b)= 1$.

Please see here for how to typeset common math expressions with MathJax, and see here for how to use Markdown formatting. — Zev Chonoles, Jun 18 '13 at 18:49
The approximate value, for $n$ large, is $6n^2/\pi^2$. I don't think there is a closed formula for a particular $n$. — Thomas Andrews, Jun 18 '13 at 18:53
Take a look at http://oeis.org/A018805 for formulas and such. I agree with Thomas that there seems to be no closed formula for a particular n. — Matthew Conroy, Jun 18 '13 at 19:18

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

This is essentially the same question I recently answered here, since your quantity is $$ -1+2\sum_{k=1}^n\varphi(k)=\sum_{k=1}^n\mu(k)\left\lfloor\frac nk\right\rfloor^2. $$

edited Apr 13 '17 at 12:20

Community

1

answered Jun 18 '13 at 19:54

Charles

32,999

Number of ordered pairs of coprime integers from $1$ to $N$

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