The difference between "当たり" and "につき" in terms of the meaning "every ~"


As mentioned at the title, I would like to ask the if there is any difference
or nuance between "当たり" and "につき" in terms of the meaning "every ~". Any case
that they cannot be used interchangeably?

For instance, do the following sentences (if they are grammatically correct)
convey the exact same meaning?

> 卵は一人 **につき** 二つまで ＝ 卵は一人 **あたり** 二つまで
>
> 一ヶ月 **につき** 一回東京へ行く ＝ 一ヶ月 **あたり** 一回東京へ行く




[There is a page](http://naniga-chigauno.st042.net/z425.html) that tries to
describe the difference, but につき and あたり(の) seemed interchangeable to me for
all the examples in the page :) So I think the difference is very small and
they are basically interchangeable. Still, there are small differences:

  * ～につき sounds more formal and stiff.
  * ～あたり tends to be used in relation to a total amount. In other words, if you did a mathematical division in your mind, ～あたり may be used more often. If I heard 1か月あたり1回(東京へ行く) instead of 毎月 or 月に1回, I may feel you have a big project in Tokyo.




[There is a page](http://naniga-chigauno.st042.net/z425.html) that tries to
describe the difference, but につき and あたり(の) seemed interchangeable to me for
all the examples in the page :) So I think the difference is very small and
they are basically interchangeable. Still, there are small differences:

  * ～につき sounds more formal and stiff.
  * ～あたり tends to be used in relation to a total amount. In other words, if you did a mathematical division in your mind, ～あたり may be used more often. If I heard 1か月あたり1回(東京へ行く) instead of 毎月 or 月に1回, I may feel you have a big project in Tokyo.

