2019

Let’s analyse the relationship between n and f(n).
Notice the outputs 0, 0, 2, 0, 2, ?, 2, 5.
The inputs are 1, 2, 3, 4, 5, 6, 7, 8.
f(1)=0, f(2)=0, f(4)=0. These inputs (1, 2, 4) are powers of 2 ($1=2^0$, $2=2^1$, $4=2^2$).
f(3)=2, f(5)=2, f(7)=2. These inputs (3, 5, 7) are primes greater than 2.
f(8)=5. This input (8) is $2^3$.
f(6)=x. This input (6) is not a power of 2 and not a prime.

Let’s hypothesize a rule based on the observed pattern:
– If n is a power of 2, $n=2^k$: the output depends on k? f(1)=f($2^0$)=0, f(2)=f($2^1$)=0, f(4)=f($2^2$)=0, f(8)=f($2^3$)=5.
– If n is a prime greater than 2: f(n)=2. (Matches f(3), f(5), f(7)).
– If n is composite and not a power of 2: f(n)=? (Only n=6 in the given range).

Let’s refine the rule based on inputs being powers of 2 vs not powers of 2.
– If n is a power of 2: $n=2^k$. f(1)=0, f(2)=0, f(4)=0, f(8)=5. This still doesn’t follow a simple rule like k, k+c, k^2, etc.
– If n is NOT a power of 2: f(3)=2, f(5)=2, f(7)=2. This strongly suggests f(n)=2 for n which are not powers of 2, with a potential exception for n=8.

Let’s assume the rule is:
f(n) = 2 if n is not a power of 2.
f(n) = 0 if n is a power of 2, except for n=8.
f(8) = 5.

Let’s test this rule:
f(1) = f($2^0$). Power of 2, not 8. Rule says 0. Matches.
f(2) = f($2^1$). Power of 2, not 8. Rule says 0. Matches.
f(3). Not a power of 2. Rule says 2. Matches.
f(4) = f($2^2$). Power of 2, not 8. Rule says 0. Matches.
f(5). Not a power of 2. Rule says 2. Matches.
f(6). Not a power of 2 ($6 = 2 \times 3$). Rule says 2. So x=2.
f(7). Not a power of 2. Rule says 2. Matches.
f(8) = f($2^3$). This is the special case $n=8$. Rule says 5. Matches.

This rule consistently explains the sequence values 0, 0, 2, 0, 2, ?, 2, 5, and predicts x=2.
The number 6 is not a power of 2 ($1, 2, 4, 8, 16, …$). According to the rule, f(6) should be 2.
Therefore, x = 2.

Let’s check the numbers between 200 and 230:
– 201: Divisible by 3 (sum of digits 2+0+1=3). Not prime.
– 202: Divisible by 2. Not prime.
– 203: $203 = 7 \times 29$. Not prime.
– 204: Divisible by 2. Not prime.
– 205: Divisible by 5. Not prime.
– 206: Divisible by 2. Not prime.
– 207: Divisible by 3 (sum of digits 2+0+7=9). Not prime.
– 208: Divisible by 2. Not prime.
– 209: $209 = 11 \times 19$. Not prime.
– 210: Divisible by 10. Not prime.
– 211: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $14^2 = 196$, $15^2 = 225$. $\sqrt{211} \approx 14.5$. Primes to check up to 13. 211 is prime.
– 212: Divisible by 2. Not prime.
– 213: Divisible by 3 (sum of digits 2+1+3=6). Not prime.
– 214: Divisible by 2. Not prime.
– 215: Divisible by 5. Not prime.
– 216: Divisible by 2. Not prime.
– 217: $217 = 7 \times 31$. Not prime.
– 218: Divisible by 2. Not prime.
– 219: Divisible by 3 (sum of digits 2+1+9=12). Not prime.
– 220: Divisible by 10. Not prime.
– 221: $221 = 13 \times 17$. Not prime.
– 222: Divisible by 2. Not prime.
– 223: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{223} \approx 14.9$. Primes to check up to 13. 223 is prime.
– 224: Divisible by 2. Not prime.
– 225: Divisible by 5. Not prime.
– 226: Divisible by 2. Not prime.
– 227: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{227} \approx 15.07$. Primes to check up to 13. 227 is prime.
– 228: Divisible by 2. Not prime.
– 229: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{229} \approx 15.13$. Primes to check up to 13. 229 is prime.

The prime numbers between 200 and 230 are 211, 223, 227, and 229.
There are 4 prime numbers in this range.