Indivisibility of class numbers of real quadratic fields

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the 'Cohen and Lenstra Heuristics' give a probability that p inverted iota h(D). If p > 3 is prime, then subject to a mild condition, we show that #{0 < D < X\p inverted iota h(D)} >>p root X/log X. This condition holds for each 3 < p < 5000.