The nth partial sum of the Maclaurin series for ea/b, where a and b are integers, becomes an integer when multiplied by n!bn. This integer is related to many combinatorial properties of interest and is also directly tied to an exact computation of n, a b , where is the incomplete gamma function. This paper presents very short formulas that give this integer exactly when |a| ? 2. For larger a the method extends and while not as fast as the smaller cases, it is an improvement on existing computational methods. The cases a = ±1 were known for many b-values. The approach here extends the general idea to all rationals by making use of a congruence to overcome the error inherent in the truncation of the Maclaurin series. A side-e?ect of the investigation is a new analytic lower bound on the number of times a prime a appears in a factorial: n a1 loga(n + 1).
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