Back in those early days of space exploration, the manned missions were formally Soviet missions, since Russia was only one of 15 socialistic republics of USSR. Korolev himself was born in Ukrainian SSR and started his studying in Kiev. As a side note, I myself had a priviledge to study at Kiev Politechnical University in that same room Korolev had been studiyng. Only after collapse of Soviet Union in 1991 the missions started to be Russian.
This source (in Russian) lists all manned missions of Soviet union. The numbering is sequential throughout the respective space program, i.e. Vostok-xx; Voskhod-xx, Soyuz-xx etc.
For later modifications of Soyuz spacecraft, the mission name was reflecting the version of the spacecraft: Soyuz T-xx; Soyuz TM-xx.
After Soviet Union has collapsed in 1991, all further missions, starting from Soyuz TM-13 were indeed Russian: Soyuz TM-xx, Soyuz TMA-xx, Soyuz MC-xx. This source lists all Russian manned missions (i.e. since 1991 to date).
Actually, TM-13 took off from Soviet Union and landed in Russia. Krikalev had to spend an unexpected prolonged time on orbit due to SU collapse (akin to Tom Hanks' character in movie Terminal)
It is not an unknown historical fact that in Soviet union, when some early missions failed, the sequential number was not assigned to that mission. This was done with the aim to keep the (essentially, faked) "spotless" launch history.
Later they started to admit failures and this was to an extent reflected in missions numbering system.
According to the above sources, there are duplicates in sequential numbering of some missions, and some missions numbers are omitted, but the reasons for that need to be looked into on case by case basis. This would be too big of an effort for this answer here.
Also some missions, such as T's have started before previous versions of Soyuz were done flying, so historical timeline contains a mixup of the names, for example, Soyuz-36, then Soyuz T-2, then Soyuz-37 etc. Also, in the days of Russia, they for whatever reason brought back TM's after TMA-10 and restarted from 11 (i.e. TM-11), this gave double naming for TM-11, TM-12 and TM-13 (i.e. same name for separate different missions).
Hopefully the sequential list in the two links above should be enough for this answer.