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See this oeis entry. This can be easily reduced to Ramanujan-Nagell problem. It was conjectured by Ramanujan in 1913 and resolved by Nagell in 1948 (where the year 1930 come from is not clear).


Plato does not define perfect numbers anywhere. The earliest extant definition is in Euclid VII, def. 22. Although Plato uses the term "perfect number" I do not think it certain that he understood this phrase in the same way as Euclid.


I think that this is what you are asking for: Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différents problèmes, Journal de l'École polytechnique, $18^e$ cahier, 11 (1820), p. 417–489. Here you can find the scan of the original paper (in French, of course)...

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