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The series of Armenian stamps known as "First Yessayan" were only issued in 1922 - 23 with overprints. However, since the overprints were all applied with single handstamps or in manuscript, it is inevitable that some stamps got missed. One should expect to find the occasional used stamp without an overprint.
In addition, both the Gibbons catalogue (following Tchilingirian and Ashford) and the Michel catalogue (following Zakiyan and Saltykov) list deliberate use of two values without surcharge.
Gibbons says that the 250 rouble perforated stamp was used without Manuscript surcharge " 1 k" ( in red or violet) at three post offices in March - May 1923. It lists those post offices as Delizhan, Karaklis and Keshikend.
Above are three used examples of the 250 r perforated stamp. Stefan Berger has studied all three of them with his microscope, telescope and goodness knows what else and can find no visible trace of a surcharge. The third stamp above has a readable ALEXANDROPOL cancel and I think the other two are also with Alexadropol cancels - the similarities are obvious. The first stamp in the row has an Agathon Faberge acquisition note from 1927, the second a MAISON ROMEKO mark. All three seem genuine in all respects to me and to Stefan Berger. So it looks like that at Alexandropol they also used this stamp without surcharge and probably deliberately since here we have three examples, not just a random one from a missed surcharge.
Michel says that the 25 rouble ( unspecified whether perforated or imperforate or both) which was later surcharged as a 4 kopeck stamp was also issued unsurcharged for use as a 1500 rouble stamp in the period January - May 1922. [This paragraph rewritten on the basis of a Comment from Alexander Epstein ]
Anyway, the 25 rouble shown above with ERIVAN cancel of November 1922 does not appear to have any regular 4 kop surcharge. However, and just to complicate matters, there is a small violet ink mark under the day of the date line in the cancellation. This could either be randon or it could be a squiggle representing a number "4". Maybe this is a stamp from which the regular surcharge was accidentally omitted and a clerk made a manuscript correction. In the absence of other examples, who knows? All that is clear is that this stamp also lacks a regular surcharge.