How many numbers 10 through 87 have the sum of their digits equal to a perfect square?

Numbers whose digits add to a perfect square between 10 and 87 are:

10 = 1 + 0 = 1 = 12
13 = 1 + 3 = 4 = 22
18 = 1 + 8 = 9 = 32
22 = 2 + 2 = 4 = 22
27 = 2 + 7 = 9 = 32
31 = 3 + 1 = 4 = 22
36 = 3 + 6 = 9 = 32
40 = 4 + 0 = 4 = 22
45 = 4 + 5 = 9 = 32
54 = 5 + 4 = 9 = 32
63 = 6 + 3 = 9 = 32
72 = 7 + 2 = 9 = 32
81 = 8 + 1 = 9 = 32

This is a total of 13 numbers.