Balti Tee 2010 · Ülesanne 16

The answer is $9$. For every $k$ we have $s(k) \equiv k \pmod{9}$. Calculating remainders modulo $9$ we have the following table

$m$	0	1	2	3	4	5	6	7	8
$m^2$	0	1	4	0	7	7	0	4	1
$m^6$	0	1	1	0	1	1	0	1	1

If $d(k) = 3$, then $k = p^2$ with $p$ a prime, but $p^2 \equiv 3 \pmod{9}$ is impossible. This shows that $3$ is not an amusing number. If $d(k) = 5$, then $k = p^4$ with $p$ a prime, but $p^4 \equiv 5 \pmod{9}$ is impossible. This shows that $5$ is not an amusing number. If $d(k) = 7$, then $k = p^6$ with $p$ a prime, but $p^6 \equiv 7 \pmod{9}$ is impossible. This shows that $7$ is not an amusing number. To see that $9$ is amusing, note that $d(36) = s(36) = 9$.