Abstract
The Pseudo-Smarandache Function is part of number theory. The function comes from the Smarandache Function. The Pseudo-Smarandache Function is represented by Z(n) where n represents any natural number. The value for a given Z(n) is the smallest integer such that 1+2+3+... + Z(n) is divisible by n. Within the Pseudo-Smarandache Function, there are several formulas which make it easier to find the Z(n) values.Formulas have been developed for most numbers including: a) p, where p equals a prime number greater than two; b) b, where p equals a prime number, x equals a natural number, and b=-px; c) x, where x equals a natural number, if x/2 equals an odd number greater than two; d) x, where x equals a natural number, if x/3 equals a prime number greater than three. Therefore, formulas exist in the Pseudo-Smarandache Function for all values of b except for the following: a) x, where x = a natural number, if x/3 = a nonprime number whose factorization is not 3x; b) multiples of four that are not powers of two. All of these formulas are proven, and their use greatly reduces the effort needed to find Z(n) values.
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