Only seven such numbers are known. Referring back to A204539, we cannot be certain that these are the only numbers with the property basin(n) = 1, but there are no others for n less than 12,000,000. Furthermore the average value of basin(n) is approximately equal to n/3 for large values of n, which makes it extremely unlikely that another one will ever be found.
For unknown reasons, all integers >5 with basin equal to 1 or 2 are even. See A185001 for the list of known numbers with basin = 2.