with the occasion of

In my mother tongue, there are two different phrases. Translating them literally: “out of the occasion of…” and “with the occasion of....” I’ve found the English equivalent of the first one, it is “on the occasion”. For e.g. I wish you or give you this or that on the occasion of your birthday/our wedding anniversary/etc. It may happen that it is not on the same day, for e.g. I wish something to somebody five days earlier or later, it doesn’t matter, only the cause is important. But I haven’t found the equivalent of the other phrase, where only the time is important. As a matter of fact it is a simple emphasising of “when.” For e.g.: “That thing happened with the occasion of Fred’s visit.” (I.e. when Fred visited us.) It may also be a birthday, just like anything else, for e.g.: “I lost my cap with the occasion of Joe’s birthday, but naturally it wasn’t a gift for Joe that I lost my cap. So what is the English phrase, if it is?