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I am a beginner in music with little knowledge, but like once in a month spend a little time playing with a digital keyboard.

I noticed that if I have some notes of a song which got 4# in the beginning of the stave, I can play the song like there is 3♭ instead. The same happens if I have 4 bemols (flats) then I can play the song like it has 3 diesis (sharps) (yes, it will sound a bit higher, but not that different).

I was curious if other combinations exists, let's say we have a song in 2 bemols (flats), what is the equivalence of it in diesis (sharps)? I couldn't find it myself. Is there a name for this phenomenon so I can learn more?

I'm not aware of a name for this phenomenon, it's just a quick way to transpose music based on how the tonal system works out.

In short, when you're in a key, look at the key signature. Take the number of accidentals in the key and replace them with the mod-7 complement of the other accidental type and you're left with a key built a half step away from the original tonic.

So you're in E major with 4 sharps. Let's take the mod-7 complement of the other accidental type: 7-4=3, so we're left with 3 flats, which is E♭ major, one half step away from the original tonic of E.

You're now asking about 2 flats in the key signature; this is B♭ major. 7-2=5, so a key of 5 sharps will be B major.

This trick is especially fun in C, which has 0 sharps or flats. The mod-7 complement of 0 is 7, so if we have 7 sharps in the key signature, we're in C♯ major; 7 flats makes it C♭ major!

Note that this trick isn't exclusive to major; it works for minor keys as well.

Lastly, know that this works perfectly until you encounter accidentals in the music; you'll have to have a more contextual understanding of those accidentals to know how they should be interpreted in your new key.

I was curious if other combinations exists, let's say we have a song in 2 bemols (flats), what is the equivalence of it in diesis (sharps)? I couldn't find it myself. Is there a name for this phenomenon so I can learn more?

There isn’t a name for this phenomen. But we can find one. In German I would call them “gleichnamige” Tonarten, in English this would be “same named” keys.
(Now, as we know they are not exactly the same name, as the “related” key*1) has added a -bemol (flat) and is a half tone lower!)

The phenomen can be explained quite simply by the circle of fifths and by the tones of the twelve
tone scale (fixed do names!)

Do, Re-bemol, Re, Mi-bemol, Mi, Fa, Fa#/Sol-bemol, Sol, La-bemol, La, Si-bemol, Si.

Of each tone of the doremi scale exist two keys: one on the -bemol site (flats) and the -diesis (sharps) site. The two keys with the same name are differing logically a minor second respectively 7 fifths
and can be played by exchanging the amount of # with the amount of b of its same named (“related”) key:

So this “related” keys in question are:

Si-bemol (2bemol) and Si (5#)

analogically we get (starting with Re-bemol):

Re-bemol and Re

La-bemol and La

Mi-bemol and Mi

Si-bemol and Si

Fa and Fa#

Do and Do# (Re-bemol)

Sol and Sol# (La-bemol)

This is a consequence of the key signatures. Basically, what you're doing is changing the number of accidentals in the key signature by seven (some would argue that's an oxymoron, but you all know what I mean). It turns out that by doing this, you've transposed the song into the key a half-step up. If you want the same exact key, you can add or subtract 12, but that makes for some ugly key signatures.

It's pretty intuitive; change the number of accidentals by seven, and you lower or raise every note by a half-step. That's the definition of how to transpose by a half-step!