All Solutions of the Equation

The number of divisors function den), is a classic function of number theory, having been defined centuries ago. In contrast, the Smarandache function Sen), was defined only a few decades ago. The purpose of this paper is to tind all solutions to a simple equation involving both functions. Theorem: The only solutions to the equation Sen) + den) = n, n > 0 are 1, 8 and 9. Proof: Since S( 1) = 0 and d( 1) = 1 we have verified the special case of n = 1. Furthermore, with S(P) = p for p a prime, it follows that any solution must be composite. The following results are well-known. a) d(Pll ... Pkk ) = (a1 + 1) ... (ak -+1) b) S(pk) :::; kp c) S(Pll ... Pkk) = max { S(Pll) ... S(Pkk) }