Too happy

Clearly all one digit numbers are 2-happy.

For a $k$ -digit number, twice the sum of the digits is at most $18k$ . When $k \geq 3$ , we have $18k < 10^{k-1}$ . Thus, for $k \geq 3$ , all $k$ -digit numbers have value greater than twice their digit sum.

For 2-digit numbers, we make two observations: 1) If $\overline{ab}$ is not 2-happy then $\overline{ac}$ is not 2-happy for $c < b$ . 2) If $x$ is not 2-happy, then $x+10$ is not 2-happy. We next observe that $29$ and $18$ are not 2-happy. Combining these facts with the above 2 observations, we have that the only possible 2-happy 2-digit number is $19$ . We verify that this number is in fact 2-happy, since $(1 + 9) \times 2 = 20 > 19$ .

Thus, there are $10$ 2-happy numbers $1,2,3,4,5,6,7,8,9,19$ .