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The Normal Behavior of the Smarandache Function

By Kevin Ford


Let S(n) be the smallest integer k so that n|k!. This is known as the Smarandache function and has been studied by many authors. If P (n) denotes the largest prime factor of n, it is clear that S(n) � P (n). In fact, S(n) = P (n) for most n, as noted by Erdös [E]. This means that the number, N(x), of n � x for which S(n) � = P (n) is o(x). In this note we prove an asymptotic formula for N(x). First, denote by ρ(u) the Dickman function, defined by � u ρ(u) = 1 (0 � u � 1), ρ(u) = 1 − For u> 1 let ξ = ξ(u) be defined by It can be easily shown that u = eξ − 1 ξ ξ(u) = log u + log 2 u +

Year: 2008
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