The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

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This note is concerned with the number C(n) of ordered non-trivial factorizations of an integer n in the special case where n is square free. If F (m) denotes C(p1…Pm) where pi. are distinct primes, it is shown thatand that