점화 조건 함수 최댓값

[출제의도] 함수의 뜻 이해하기 $f ( 1 ) = 1$ $f ( 2 ) = f ( 1 ) = 1$ $f ( 3 ) = f ( 1 ) + 1 = 2$ $f ( 4 ) = f ( 2 ) = f ( 1 ) = 1$ $f ( 5 ) = f ( 2 ) + 1 = f ( 1 ) + 1 = 2$ $f ( 6 ) = f ( 3 ) = f ( 1 ) + 1 = 2$ $f ( 7 ) = f ( 3 ) + 1 = f ( 1 ) + 1 + 1 = 3$ $f ( 8 ) = f ( 4 ) = f ( 2 ) = f ( 1 ) = 1$ $f ( 9 ) = f ( 4 ) + 1 = f ( 2 ) + 1 = f ( 1 ) + 1 = 2$ $f ( 10 ) = f ( 5 ) = f ( 2 ) + 1 = f ( 1 ) + 1 = 2$ $\vdots$ $\begin{aligned} f ( 128 ) & = f ( 64 ) = f ( 32 ) = f ( 16 ) \\ & = f ( 8 ) = f ( 4 ) = f ( 2 ) = f ( 1 ) = 1 \end{aligned}$ $1 \leq n \leq 128$인 $n$에서 $f ( 1 ) = 1$ $f ( 3 ) = f ( 1 ) + 1 = 2$ $f ( 7 ) = f ( 3 ) + 1 = 3$ $f ( 15 ) = f ( 7 ) + 1 = 4$ $f ( 31 ) = f ( 15 ) + 1 = 5$ $f ( 63 ) = f ( 31 ) + 1 = 6$ $f ( 127 ) = f ( 63 ) + 1$ $= f ( 31 ) + 1 + 1$ $= f ( 15 ) + 1 + 1 + 1$ $= f ( 7 ) + 1 + 1 + 1 + 1$ $= f ( 3 ) + 1 + 1 + 1 + 1 + 1$ $= f ( 1 ) + 1 + 1 + 1 + 1 + 1 + 1$ $= 7$ $f ( n )$은 $n = 127$일 때 최댓값 $7$을 갖는다. 따라서 $a + M = 127 + 7 = 134$