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These occur since 1 = 10/10, 2 = 10/5, and 5 = 10/2. Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. A perfect number is defined to be one which is equal to the sum of its aliquot parts.
Euler showed that an odd perfect number, if it exists, must be of the form. (1) where is a prime of the form (Fermat's 4n+1 theorem; Burton 1989), a result similar to that derived by Frenicle in 1657 (Dickson 2005, pp. 14 and 19). In other words, an odd perfect number must be of the form.