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The explanation seems to be related to the attempted spelling of the Arabic for Osman, which came out as utman or Uthman. From wiktionary.org
From Middle French Ottoman, from post-classical Latin Ottomanus, from Ottoman Turkish عثمان, from Arabic personal name عُثْمَان (ʿuṯmān). Osman is the Turkish spelling of the male Arabic given name Uthman, therefore the Ottoman Empire is sometimes referred to as the Osman Empire, Osmanic Empire, or Osmanian Empire, after Osman I.
Its also has been discussed here on the English Language & Usage Stack.
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There is a very strong theory about Osmans real name being Ataman, a Turkic name and that his descendant Murad II(in whose reign the first historical records were written since the foundation)made historians record it as Osman in order to give the state an islamic feeling.
But of course, even if it is true, the British were absolutely unaware of this. So, as some others stated, it is probably taken from the French who recorded it based on the arabic spelling Uthman, and from there, it became Ottoman in English.
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As an Arabic speaker, Osman and Uthman/Othman are really the same name (both are Arabic/Turkish names). So calling the nation the Ottoman Empire makes sense when comparing with the name Othman.