First, I need to understand what these functions are doing. There are three functions: f2, f1, and f0. The main part runs a loop from x=0 to 31 and prints f0(x). So I need to compute f0 for each x.

Let's start with f2. f2 takes a parameter X. Inside, it initializes x and i to 0. Then it enters a while loop that continues as long as i is less than or equal to max(X,0). Since X is from the range, and x in range(32) is passed to f0, which calls f1 and f2, so X might be an integer.

In f2, while i <= max(X,0). If X is negative, max(X,0) is 0, so i <=0, so the loop might run only if i=0, but let's see.

But in the context, when f0 is called with x from 0 to 31, which are non-negative, so max(X,0) = X for X>=0.

So for f2, with X>=0, i <= X.

Now, inside the loop, there's an if condition: if (x if (x % (2 * (2 + 2))) <= 0 else 0) <= 0. This looks messy. Let's simplify it.

2 * (2 + 2) = 2 * 4 = 8. So x % 8. Then (x % 8) <= 0? But modulo operation: x % 8 is always between 0 and 7, so it's never less than 0, but it can be equal to 0. So (x % 8) <= 0 is true only if x % 8 == 0.

But there's a ternary operator: (x if (x % 8) <= 0 else 0) <= 0. So if x % 8 == 0, then it takes x, and checks if x <= 0? Else it takes 0, and checks if 0 <= 0, which is true.

So let's rewrite the condition:

If x % 8 == 0, then we check if x <= 0. But x is increasing from 0, so for x=0, x<=0 is true. For x=8, x=8>0, so false. For x=16, false, etc.

If x % 8 != 0, then we have 0, and 0<=0 is true.

So the condition is true when either x % 8 != 0 or x % 8 == 0 and x <=0. But since x starts at 0 and increases, x<=0 only when x=0. So for x=0, condition true. For x>0, if x % 8 !=0, condition true; if x % 8 ==0, condition false because x>0.

So effectively, the condition is false only when x is a positive multiple of 8.

Now, if the condition is true, then i is incremented by 1. Otherwise, i is not incremented.

After the condition, x is always incremented by 1.

The loop continues until i > X. Since i starts at 0 and is incremented only when the condition is true, we need to find how many x values satisfy the condition until i reaches X+1? Because i <= X, so when i becomes X+1, loop stops.

The loop condition is i <= max(X,0), so for X>=0, i <= X. So when i becomes X+1, loop stops.

In the loop, i is incremented only when condition is true. So we need to count the number of times condition is true until i reaches X+1? But i is incremented each time condition is true, so we need to have condition true exactly X+1 times? Let's see.

At start, i=0, x=0. We enter loop since i<=X.

We check condition: for x=0, condition true, so i incremented to 1.

Then x incremented to 1.

Now i=1, if X=0, but i=1 and X=0, then i<=X? i=1, X=0, max(0,0)=0, i=1>0, so loop would not enter? But in the code, while i <= max(X,0), so for X=0, max(0,0)=0, so i<=0? i=0 initially, so it enters, but after first iteration, i=1, which is not <=0, so loop stops. So for X=0, the loop runs only once? Let's see.

In the loop, after i is incremented, we check the condition at the beginning of next iteration. So for X=0, i=0 <=0, so enter loop. Condition true for x=0, i incremented to 1. Then x incremented to 1. Then back to while condition: i=1 <=0? No, so loop exits. Then return x-1 =1-1=0.

So for X=0, f2(0)=0.

Now, what is f2 computing? Essentially, we are counting the number of times the condition is true, but we need to find the x value when i reaches X+1? No.

The loop stops when i > X, and at that point, we return x-1.

But i is incremented only when condition is true, so the number of times condition is true is i, which when i becomes X+1, we stop, so we have encountered X+1 true conditions? But we return x-1, which is the last x value before incrementing after the last condition check.

Let's think carefully.

We have a sequence of x values: x=0,1,2,3,... and for each x, we check the condition. If condition true, we increment i. We stop when i > X, i.e., when i becomes X+1.

So we need to find the smallest x such that the number of true conditions from x=0 to x is exactly X+1.

But since we return x-1, it might be the x value where the (X+1)-th true condition occurs? Let's see.

After we increment i to X+1, we then increment x, and then check the while condition, which fails, so we return x-1.

So if the (X+1)-th true condition occurs at some x=a, then at that point, i becomes X+1 (since we increment i), then we set x to a+1, then we break the loop, and return x-1 = a.

So indeed, f2(X) returns the x value where the (X+1)-th true condition occurs.

But what is the true condition? As we said, condition is true for all x except positive multiples of 8.

So the condition is false only when x is multiple of 8 and x>0.

So the true conditions occur for x=0,1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,etc. So x=8,16,24, etc. are false.

Now, we need to find the x value where the (X+1)-th true condition occurs.

Since the pattern repeats every 8 numbers, but with one false every 8 numbers after x=0.

For every block of 8 numbers, from x=8k to 8k+7, there are 7 true conditions because x=8k is false.

But x=0 is true, so for x=0 to x=7, all true? x=0 true, x=1 to7 true, x=8 false, etc.

x=0: true

x=1 to7: true

x=8: false

x=9 to15: true

x=16: false

and so on.

So the number of true conditions up to x=n.

Let n be a non-negative integer. The number of true conditions from x=0 to x=n is n+1 minus the number of multiples of 8 between 0 and n inclusive? But x=0 is multiple of 8? 0/8=0, so yes, but for x=0, condition is true, so we don't want to subtract it because we only subtract positive multiples? In our condition, x=0 is true, so we should not subtract x=0.

The condition is false only for positive multiples of 8. So for x=8,16,24,... so multiples of 8 except 0.

So the number of false conditions up to x=n is the number of positive multiples of 8 less than or equal to n, i.e., floor(n/8) but since x=0 is not false, it's correct.

So number of true conditions up to x=n is (n+1) - floor(n/8)

Because there are n+1 numbers from 0 to n, and we subtract the number of positive multiples of 8, which is floor(n/8).

But is that correct? For n=7, floor(7/8)=0, so true conditions: 8-0=8? But from x=0 to x=7, there are 8 values, all true, so yes.

For n=8, floor(8/8)=1, so true conditions: 9-1=8? But from x=0 to x=8, there are 9 values, but x=8 is false, so only 8 true, so yes.

For n=9, floor(9/8)=1, so true conditions: 10-1=9? From x=0 to x=9, 10 values, x=8 false, so 9 true, yes.

So good.

Now, we want the smallest x such that the number of true conditions is X+1.

So we need find x such that (x+1) - floor(x/8) = X+1

But since we are counting from x=0, and we want the x where the (X+1)-th true occurs, so set number of true conditions equal to X+1.

So (x+1) - floor(x/8) = X+1

Thus x - floor(x/8) = X

So f2(X) returns the x such that x - floor(x/8) = X.

But from above, we have x - floor(x/8) = X.

For example, X=0, x=0, since 0 - floor(0/8)=0-0=0.

X=1, x=1, 1 - floor(1/8)=1-0=1.

X=2, x=2, similarly up to X=7, x=7.

X=8, we need x such that x - floor(x/8)=8. If x=8, 8 - floor(8/8)=8-1=7, too low. x=9, 9 - floor(9/8)=9-1=8, so x=9. So f2(8)=9.

And indeed, from true conditions: up to x=7, 8 true conditions? Number of true conditions up to x=7 is 8, so for X=8, we need the 9th true condition, which is at x=9? Since x=8 is false, so after x=7, next true is x=9? But x=0 is first true, so up to x=7, 8 true, so the 9th true is at x=9. Yes.

So f2(X) returns the solution to x - floor(x/8) = X.

But since x - floor(x/8) is essentially x minus the number of multiples of 8 less than x? floor(x/8) is the number of multiples of 8 from 1 to x? Actually, floor(x/8) is the number of multiples of 8 from 0 to x, but since x=0, floor(0/8)=0, so it includes 0, but since we subtract, and for x=0, it works, so ok.

We can think of x - floor(x/8) as the number of non-multiples of 8 from 0 to x? But not exactly, because from 0 to x, there are x+1 numbers, and multiples of 8 are floor(x/8)+1 including 0, but since we have x - floor(x/8), it doesn't directly give the count.

From above, we have that the number of true conditions up to x is x+1 - floor(x/8), which equals X+1, so x - floor(x/8) = X.

So for given X, f2(X) returns the x such that x - floor(x/8) = X.

This x can be found easily. Since x - floor(x/8) = X, and floor(x/8) = k for some k, so x = X + k, and since floor(x/8) = k, we have k = floor( (X + k)/8 ) so k ≈ X/7, but we can solve.

Note that for each X, x is roughly X, but since we skip multiples of 8, we have to add the number of skips.

But anyway, we don't need a closed form for now.

Now, let's look at f1(X).

f1 takes X and computes x = (X + X) + f2(X) = 2X + f2(X)

Then it has a for loop for y from 1 to X inclusive.

For each y, it sets x = 0 if (x % (1 + y)) <= 0 else x

Now, (x % (1+y)) <= 0? Since modulo is always non-negative, it is <=0 only if modulo equals 0. So if x is divisible by (1+y), then set x to 0, else leave x unchanged.

So for each y from 1 to X, if x is divisible by y+1, then x becomes 0.

After all iterations, return x.

So essentially, if for any y from 1 to X, x is divisible by y+1, then x becomes 0. But since it sets to 0, and then for subsequent y, if x is 0, then 0 % (1+y) = 0, so it remains 0. So if there exists any y in 1 to X such that x is divisible by y+1, then at the end x=0. Otherwise, if x is not divisible by any number from 2 to X+1, then x remains unchanged.

But note that if x is divisible by some number, it gets set to 0, and then stays 0.

So f1(X) returns 0 if x is divisible by any integer from 2 to X+1, otherwise it returns x.

But x is 2X + f2(X)

So f1(X) = 0 if 2X + f2(X) is divisible by any integer from 2 to X+1, else it returns 2X + f2(X)

Now, what is f2(X)? We have f2(X) = the number such that f2(X) - floor(f2(X)/8) = X

But since f2(X) is an integer, and for large X, f2(X) is about 8/7 X, but let's see.

Now, finally, f0(X).

f0 takes X, initializes x and i to 0. Then while i <= max(X,0), which for X>=0, i<=X.

Inside, if (1 - f1(x)) <= 0, then i is incremented.

Then x is incremented.

After loop, return x-1.

So similar to f2, we are finding the x where i becomes X+1, but i is incremented only when (1 - f1(x)) <= 0, which is true if f1(x) >= 1? Since 1 - f1(x) <= 0 implies f1(x) >= 1.

But f1(x) is either 0 or some number, so f1(x) >= 1 only if f1(x) > 0, which means that f1(x) is not zero, so from above, it means that when f1(x) is not zero, which means that 2x + f2(x) is not divisible by any number from 2 to x+1? But f1 is called with argument x, so in f1(X), X is x from f0? Let's see.

In f0, we have f1(x), so for each x, we compute f1(x), which is based on x.

So f1(x) = 0 if 2x + f2(x) is divisible by any integer from 2 to x+1, else f1(x) = 2x + f2(x)

And in f0, we increment i if f1(x) >= 1, i.e., if f1(x) ≠ 0, i.e., if 2x + f2(x) is not divisible by any integer from 2 to x+1.

Otherwise, if f1(x)=0, we do not increment i.

Then we return x-1 when i reaches X+1.

So f0(X) returns the x value where the (X+1)-th number for which 2x + f2(x) is not divisible by any integer from 2 to x+1 occurs.

In other words, we are finding the (X+1)-th x such that 2x + f2(x) is coprime with all numbers from 2 to x+1? But not exactly coprime, because it means that 2x + f2(x) has no divisors in the range 2 to x+1, which implies that 2x + f2(x) must be 1 or a prime number greater than x+1? But since 2x + f2(x) is at least 2x, and for x>0, 2x > x+1, so if it is not divisible by any number from 2 to x+1, it must be prime? Because if it were composite, it would have a divisor less than or equal to its square root, but since x+1 might be less than the square root, it might not be sufficient.

Let's think about the condition.

We require that 2x + f2(x) is not divisible by any integer d where 2 ≤ d ≤ x+1.

This means that 2x + f2(x) has no factors between 2 and x+1. Since 2x + f2(x) is likely larger than x+1, if it has no factors less than or equal to x+1, then it must be prime, because any composite number has a factor less than or equal to its square root, and if its square root is greater than x+1, then it might have factors between x+2 and sqrt, but since we are not checking those, it could be composite as long as all factors are greater than x+1.

For example, if x=5, then we require that 2*5 + f2(5) =10 + f2(5) is not divisible by 2,3,4,5,6.

What is f2(5)? From earlier, f2(X) = x such that x - floor(x/8)=X. For X=5, x=5 since 5 -0=5. So f2(5)=5. So 10+5=15. Now 15 is divisible by 3 and 5, so it would be set to 0, so f1(5)=0, so for f0, we would not increment i for x=5.

But if x=4, 2*4 + f2(4)=8 + f2(4). f2(4)=4, so 12. Divisible by 2,3,4,5? 12 div by 2,3,4, so f1=0.

x=3, 6 + f2(3)=6+3=9, divisible by 3, so 0.

x=2, 4 + f2(2)=4+2=6, divisible by 2,3, so 0.

x=1, 2 + f2(1)=2+1=3, divisible by 2? 3 not div by 2, so we check y from 1 to 1, so only y=1, check div by 1+1=2? 3 not div by 2, so f1(1)=3? So f1(1)=3, which is大于0, so for f0, when x=1, we increment i.

x=0, 2*0 + f2(0)=0 +0=0? Now f1(0): x=0, so we compute 2*0 + f2(0)=0. Then for loop y from 1 to 0? range(1,0+1) but if X=0, range(1,1) is empty? Let's see the code: for y in range(1, X+1). If X=0, range(1,1) which is empty, so no loop, so x=0 is returned. So f1(0)=0.

So for x=0, f1(0)=0, so not increment i.

So for f0, we start x=0, i=0. Check if 1-f1(0)=1-0=1>0, so not less than or equal to 0, so condition false, so i not incremented. Then x=1.

x=1, f1(1)=3, so 1-3=-2<=0, true, so increment i to 1. Then x=2.

And so on.

So for f0, we increment i only when f1(x) >=1, i.e., when 2x + f2(x) is not divisible by any number from 2 to x+1.

Now, what is 2x + f2(x)? Since f2(x) = the number such that f2(x) - floor(f2(x)/8) = x, so let n = f2(x), then n - floor(n/8) = x.

Thus 2x + f2(x) = 2x + n.

But n - floor(n/8) = x, so n = x + floor(n/8)

Thus 2x + n = 2x + x + floor(n/8) = 3x + floor(n/8)

But n is approximately 8/7 x, so floor(n/8) is approximately x/7, so 3x + x/7 = 22x/7, but roughly.

Since n = f2(x), and from the property, n is the x-th number that is not a multiple of 8? From earlier, f2(X) returns the X-th number that is not a multiple of 8? Let's see.

From the true condition, the condition is true for non-multiples of 8, and we count the true conditions, so the number of true conditions up to x is the number of non-multiples of 8 from 0 to x.

But when we want the (X+1)-th true, which is the (X+1)-th non-multiple of 8? But since x=0 is included, and non-multiple of 8 includes 0? 0 is multiple of 8, but in our condition, x=0 is true, so we consider it as non-multiple for the condition? But technically, 0 is multiple of 8, but for the condition, it is true, so in terms of counting, we count x=0 as a true, which is not a positive multiple, but it is a multiple.

To avoid confusion, let's list the non-multiples of 8: usually, non-multiples of 8 are numbers not divisible by 8, which includes 0? 0 is divisible by 8, so通常, non-multiples of 8 are numbers not divisible by 8, so 0 is divisible by 8, so it is a multiple, so non-multiples are 1,2,3,4,5,6,7,9,10,etc. But in our condition, x=0 is true, so we are including x=0 as if it were not multiple, but mathematically it is multiple.

So in the sequence of true conditions, we have all numbers that are either 0 or not divisible by 8. So it is all numbers except positive multiples of 8.

So the set of x for which condition true is {0} ∪ {x | x not divisible by 8}

But for large x, the density is 7/8.

Now, f2(X) returns the (X+1)-th number in this sequence? From earlier, when we want the (X+1)-th true, which is the (X+1)-th number in this sequence.

For example, X=0, we want the first true, which is x=0.

X=1, second true, x=1.

X=2, x=2, ..., X=7, x=7.

X=8, ninth true? Let's see: the true sequence: index:1:x=0, index2:x=1, index3:x=2, index4:x=3, index5:x=4, index6:x=5, index7:x=6, index8:x=7, index9:x=9, so for X=8, we want the ninth true, which is x=9, so f2(8)=9.

So indeed, f2(X) returns the (X+1)-th number that is either 0 or not divisible by 8. But since X+1, for X=0, first number is 0, etc.

But in terms of notation, let's define the sequence A where A[k] is the k-th number that is not a positive multiple of 8, including 0. So A[1]=0, A[2]=1, A[3]=2, ..., A[8]=7, A[9]=9, A[10]=10, etc.

Then f2(X) = A[X+1]

But from the equation, x - floor(x/8) = X, and for x=A[X+1], it satisfies that the number of true up to x is X+1, so x - floor(x/8) = X? From earlier, number of true up to x is x+1 - floor(x/8) = X+1, so x - floor(x/8) = X, so for x=A[X+1], we have x - floor(x/8) = X.

But since A[X+1] is the (X+1)-th true, then the number of true up to A[X+1] is X+1, so A[X+1] - floor(A[X+1]/8) = X? From above, number of true up to x is x+1 - floor(x/8), set to X+1, so x - floor(x/8) = X, so yes, for x=A[X+1], we have A[X+1] - floor(A[X+1]/8) = X.

So back to f1(x); f1(x) = 0 if 2x + f2(x) is divisible by some d between 2 and x+1, else 2x + f2(x).

But f2(x) = A[x+1] from above? Since f2(X) with X=x, so f2(x) = A[x+1]

So 2x + f2(x) = 2x + A[x+1]

And A[x+1] is the (x+1)-th number that is not positive multiple of 8, so A[x+1] ≈ (8/7)(x+1) but since A[1]=0, for large x, A[x+1] ≈ (8/7)x roughly.

But let's compute small values.

We need to compute f0(x) for x from 0 to 31.

Since f0(x) returns the x where the (x+1)-th occurrence of f1(x) >=1 happens, but f0(x) is called with x, which is from the outer loop, so for each x in range(32), we compute f0(x), which is the value for that x.

But from the code, for x in range(32): print(f0(x))

So we need to compute f0(0), f0(1), f0(2), ..., f0(31)

And f0(X) returns the number such that it is the (X+1)-th integer for which f1(x) >=1, i.e., for which 2x + A[x+1] is not divisible by any number from 2 to x+1.

But let's compute f2(X) for small X.

First, compute f2(X) for X from 0 to maybe 32.

From x - floor(x/8) = X.

For X=0, x=0

X=1, x=1

X=2, x=2

...
X=7, x=7

X=8, x=9 because 9 - floor(9/8)=9-1=8

X=9, x=10 because 10-1=9

X=10, x=11

X=11, x=12

X=12, x=13

X=13, x=14

X=14, x=15

X=15, x=17 because 17 - floor(17/8)=17-2=15

X=16, x=18 because 18-2=16

X=17, x=19

X=18, x=20

X=19, x=21

X=20, x=22

X=21, x=23

X=22, x=24? 24 - floor(24/8)=24-3=21, so for X=21, x=24? But X=22, need x=25? 25-3=22, yes.

X=23, x=26? 26-3=23

X=24, x=27? 27-3=24

X=25, x=28? 28-3=25

X=26, x=29? 29-3=26

X=27, x=30? 30-3=27

X=28, x=31? 31-3=28

X=29, x=33? 33-4=29

And so on.

But for f1(x), we need f2(x) for x, so for x from 0 upwards.

Now let's compute f1(x) for x from 0.

f1(x) = 2x + f2(x) then modified by the loop.

But first, compute 2x + f2(x)

x | f2(x) | 2x + f2(x)

x=0: f2(0)=0, so 0

x=1: f2(1)=1, so 2+1=3

x=2: f2(2)=2, so 4+2=6

x=3: f2(3)=3, so 6+3=9

x=4: f2(4)=4, so 8+4=12

x=5: f2(5)=5, so 10+5=15

x=6: f2(6)=6, so 12+6=18

x=7: f2(7)=7, so 14+7=21

x=8: f2(8)=9, so 16+9=25

x=9: f2(9)=10, so 18+10=28

x=10: f2(10)=11, so 20+11=31

x=11: f2(11)=12, so 22+12=34

x=12: f2(12)=13, so 24+13=37

x=13: f2(13)=14, so 26+14=40

x=14: f2(14)=15, so 28+15=43

x=15: f2(15)=17, so 30+17=47

x=16: f2(16)=18, so 32+18=50

x=17: f2(17)=19, so 34+19=53

x=18: f2(18)=20, so 36+20=56

x=19: f2(19)=21, so 38+21=59

x=20: f2(20)=22, so 40+22=62

x=21: f2(21)=23, so 42+23=65

x=22: f2(22)=24, so 44+24=68

x=23: f2(23)=25, so 46+25=71

x=24: f2(24)=26, so 48+26=74

x=25: f2(25)=27, so 50+27=77

x=26: f2(26)=28, so 52+28=80

x=27: f2(27)=29, so 54+29=83

x=28: f2(28)=30, so 56+30=86

x=29: f2(29)=31, so 58+31=89

x=30: f2(30)=32, so 60+32=92

x=31: f2(31)=33, so 62+33=95

x=32: f2(32)=34, but we may need up to x=31 for f0, but f0 might call f1 with larger x, so we need to compute further later.

Now for each x, we have to check if 2x + f2(x) is divisible by any d from 2 to x+1.

If it is, then f1(x)=0, else f1(x)=2x+f2(x)

Then for f0, we need to find for which x, f1(x) !=0, and then find the (X+1)-th such x.

Now let's compute for small x.

x=0: 2x+f2(x)=0. Now check divisors from 2 to 0+1=1? range(1,1) empty, so no check, so f1(0)=0. So for f0, not increment i.

x=1: 2*1 + f2(1)=2+1=3. Check if divisible by d from 2 to 2? only d=2. 3 not div by 2, so f1(1)=3. So for f0, increment i.

x=2: 4+2=6. Check d from 2 to 3? d=2,3. 6 div by 2, so set to 0. So f1(2)=0.

x=3: 6+3=9. Check d from 2 to 4? d=2,3,4. 9 div by 3, so 0.

x=4: 8+4=12. Check d from 2 to 5? div by 2,3,4, so 0.

x=5: 10+5=15. Check d from 2 to 6? div by 3,5, so 0.

x=6: 12+6=18. Check d from 2 to 7? div by 2,3,6, so 0.

x=7: 14+7=21. Check d from 2 to 8? div by 3,7, so 0.

x=8: 16+9=25. Check d from 2 to 9? 25 is div by 5? 5 is between 2 and 9, yes, so div by 5, so 0.

x=9: 18+10=28. Check d from 2 to 10? div by 2,4,7, so 0.

x=10: 20+11=31. Check d from 2 to 11? 31 is prime, so not div by any d from 2 to 11? 31 is greater than 11, and since it is prime, no divisors, so f1(10)=31. So increment i for f0.

x=11: 22+12=34. Check d from 2 to 12? 34 div by 2,17, so since div by 2, so 0.

x=12: 24+13=37. Check d from 2 to 13? 37 is prime, so not div by any d in 2-13, so f1(12)=37. increment i.

x=13: 26+14=40. Check d from 2 to 14? div by 2,4,5,8,10, so 0.

x=14: 28+15=43. Check d from 2 to 15? 43 is prime, so not div, so f1(14)=43. increment i.

x=15: 30+17=47. Check d from 2 to 16? 47 is prime, so f1(15)=47. increment i.

x=16: 32+18=50. Check d from 2 to 17? div by 2,5,10, so 0.

x=17: 34+19=53. Check d from 2 to 18? 53 is prime, so f1(17)=53. increment i.

x=18: 36+20=56. Check d from 2 to 19? div by 2,4,7,8,14, so 0.

x=19: 38+21=59. Check d from 2 to 20? 59 is prime, so f1(19)=59. increment i.

x=20: 40+22=62. Check d from 2 to 21? div by 2,31, so since div by 2, so 0.

x=21: 42+23=65. Check d from 2 to 22? div by 5,13, so 5 and 13 are between 2 and 22, so div, so 0.

x=22: 44+24=68. Check d from 2 to 23? div by 2,4,17, so 0.

x=23: 46+25=71. Check d from 2 to 24? 71 is prime, so f1(23)=71. increment i.

x=24: 48+26=74. Check d from 2 to 25? div by 2,37, so 0.

x=25: 50+27=77. Check d from 2 to 26? div by 7,11, so 0.

x=26: 52+28=80. Check d from 2 to 27? div by 2,4,5,8,10,16,20, so 0.

x=27: 54+29=83. Check d from 2 to 28? 83 is prime, so f1(27)=83. increment i.

x=28: 56+30=86. Check d from 2 to 29? div by 2,43, so 0.

x=29: 58+31=89. Check d from 2 to 30? 89 is prime, so f1(29)=89. increment i.

x=30: 60+32=92. Check d from 2 to 31? div by 2,4,23,46, so 0.

x=31: 62+33=95. Check d from 2 to 32? div by 5,19, so 0.

x=32: f2(32)=? From earlier, f2(X) for X=32, need x such that x - floor(x/8)=32. Since x=32, 32-4=28, too low. x=33,33-4=29, x=34,34-4=30, x=35,35-4=31, x=36,36-4=32, so f2(32)=36. So 2*32 +36=64+36=100. Check d from 2 to 33? div by 2,4,5,10,20,25,50, so 0.

x=33: f2(33)=? x - floor(x/8)=33. x=37? 37-4=33, so f2(33)=37. 2*33+37=66+37=103. Check d from 2 to 34? 103 is prime, so f1(33)=103. increment i.

And so on.

Now for f0, we need to find the sequence where f1(x) !=0, i.e., where we increment i.

From above, for x=1, f1=3, so increment

x=10, f1=31, increment

x=12, f1=37, increment

x=14, f1=43, increment

x=15, f1=47, increment

x=17, f1=53, increment

x=19, f1=59, increment

x=23, f1=71, increment

x=27, f1=83, increment

x=29, f1=89, increment

x=33, f1=103, increment

etc.

Now f0(X) returns the (X+1)-th such x. That is, the (X+1)-th x where f1(x) !=0.

But f0 is called with X, and it returns the x value.

So for example, f0(0) should return the first x where f1(x) !=0. From above, the first x is x=1.

f0(1) should return the second x, which is x=10.

f0(2) should return the third x, which is x=12.

And so on.

Now we need to compute for X from 0 to 31.

But we need to find enough x where f1(x) !=0 to get up to the 32nd such x? Since X up to 31, we need the (31+1)=32nd x where f1(x) !=0.

From the list above, we have only up to x=29 for now, so we need to compute further.

Let's list the x where f1(x) !=0.

From above:

x=1: f1=3

x=10:31

x=12:37

x=14:43

x=15:47

x=17:53

x=19:59

x=23:71

x=27:83

x=29:89

x=33:103

x=34? Let's compute x=34.

x=34: f2(34)=? First, find f2(34). We need n such that n - floor(n/8)=34.

n=34, floor(34/8)=4, 34-4=30<34

n=35,35-4=31

n=36,36-4=32

n=37,37-4=33

n=38,38-4=34? 38-4=34, yes. So f2(34)=38.

Then 2*34 +38=68+38=106.

Check if divisible by any d from 2 to 35? 106 div by 2,53, so since div by 2, so f1(34)=0.

x=35: f2(35)=? n - floor(n/8)=35. n=39? 39-4=35, yes. So f2(35)=39.

2*35+39=70+39=109.

Check d from 2 to 36? 109 is prime? 109 is prime, so not div, so f1(35)=109. So increment i.

So x=35: f1=109

x=36: f2(36)=? n - floor(n/8)=36. n=40? 40-5=35? floor(40/8)=5, 40-5=35<36. n=41,41-5=36, yes. So f2(36)=41.

2*36+41=72+41=113.

113 is prime, so f1(36)=113. increment.

x=37: f2(37)=? n - floor(n/8)=37. n=42? 42-5=37, yes. So f2(37)=42.

2*37+42=74+42=116.

116 div by 2,4,29,58, so div by 2, so f1(37)=0.

x=38: f2(38)=? n - floor(n/8)=38. n=43? 43-5=38, yes. So f2(38)=43.

2*38+43=76+43=119.

119 div by 7,17, so since 7 between 2 and 39, so div, so f1(38)=0.

x=39: f2(39)=? n - floor(n/8)=39. n=44? 44-5=39, yes. So f2(39)=44.

2*39+44=78+44=122.

122 div by 2,61, so 0.

x=40: f2(40)=? n - floor(n/8)=40. n=45? 45-5=40, yes. So f2(40)=45.

2*40+45=80+45=125.

125 div by 5,25, so 5 between 2 and 41, so 0.

x=41: f2(41)=? n - floor(n/8)=41. n=46? 46-5=41, yes. So f2(41)=46.

2*41+46=82+46=128.

128 div by 2,4,8,16,32,64, so 0.

x=42: f2(42)=? n - floor(n/8)=42. n=47? 47-5=42, yes. So f2(42)=47.

2*42+47=84+47=131.

131 is prime, so f1(42)=131. increment.

So x=42: f1=131

x=43: f2(43)=? n - floor(n/8)=43. n=48? 48-6=42? floor(48/8)=6,48-6=42<43. n=49,49-6=43, yes. So f2(43)=49.

2*43+49=86+49=135.

135 div by 3,5,9,15,27,45, so 0.

x=44: f2(44)=? n - floor(n/8)=44. n=50? 50-6=44, yes. So f2(44)=50.

2*44+50=88+50=138.

138 div by 2,3,6,23,46,69, so 0.

x=45: f2(45)=? n - floor(n/8)=45. n=51? 51-6=45, yes. So f2(45)=51.

2*45+51=90+51=141.

141 div by 3,47, so 0.

x=46: f2(46)=? n - floor(n/8)=46. n=52? 52-6=46, yes. So f2(46)=52.

2*46+52=92+52=144.

144 div by 2,3,4,6,8,9,12,16,18,24,36,48,72, so 0.

x=47: f2(47)=? n - floor(n/8)=47. n=53? 53-6=47, yes. So f2(47)=53.

2*47+53=94+53=147.

147 div by 3,7,21,49, so 0.

x=48: f2(48)=? n - floor(n/8)=48. n=54? 54-6=48, yes. So f2(48)=54.

2*48+54=96+54=150.

150 div by 2,3,5,6,10,15,25,30,50,75, so 0.

x=49: f2(49)=? n - floor(n/8)=49. n=55? 55-6=49, yes. So f2(49)=55.

2*49+55=98+55=153.

153 div by 3,9,17,51, so 0.

x=50: f2(50)=? n - floor(n/8)=50. n=56? 56-7=49? floor(56/8)=7,56-7=49<50. n=57,57-7=50, yes. So f2(50)=57.

2*50+57=100+57=157.

157 is prime, so f1(50)=157. increment.

So x=50: f1=157

And so on. We need to find the sequence until we have 32 values.

Now let's list all x where f1(x) !=0, i.e., where we have increment.

From above:

1,10,12,14,15,17,19,23,27,29,35,36,42,50,...

We need more.

x=51: f2(51)=? n - floor(n/8)=51. n=58? 58-7=51, yes. So f2(51)=58.

2*51+58=102+58=160.

160 div by 2,4,5,8,10,16,20,32,40,80, so 0.

x=52: f2(52)=? n - floor(n/8)=52. n=59? 59-7=52, yes. So f2(52)=59.

2*52+59=104+59=163.

163 is prime, so f1(52)=163. increment.

So x=52: f1=163

x=53: f2(53)=? n - floor(n/8)=53. n=60? 60-7=53, yes. So f2(53)=60.

2*53+60=106+60=166.

166 div by 2,83, so 0.

x=54: f2(54)=? n - floor(n/8)=54. n=61? 61-7=54, yes. So f2(54)=61.

2*54+61=108+61=169.

169 div by 13, so 13 between 2 and 55, so 0.

x=55: f2(55)=? n - floor(n/8)=55. n=62? 62-7=55, yes. So f2(55)=62.

2*55+62=110+62=172.

172 div by 2,4,43,86, so 0.

x=56: f2(56)=? n - floor(n/8)=56. n=63? 63-7=56, yes. So f2(56)=63.

2*56+63=112+63=175.

175 div by 5,7,25,35, so 0.

x=57: f2(57)=? n - floor(n/8)=57. n=64? 64-8=56? floor(64/8)=8,64-8=56<57. n=65,65-8=57, yes. So f2(57)=65.

2*57+65=114+65=179.

179 is prime, so f1(57)=179. increment.

So x=57: f1=179

x=58: f2(58)=? n - floor(n/8)=58. n=66? 66-8=58, yes. So f2(58)=66.

2*58+66=116+66=182.

182 div by 2,7,13,14,26,91, so 0.

x=59: f2(59)=? n - floor(n/8)=59. n=67? 67-8=59, yes. So f2(59)=67.

2*59+67=118+67=185.

185 div by 5,37, so 0.

x=60: f2(60)=? n - floor(n/8)=60. n=68? 68-8=60, yes. So f2(60)=68.

2*60+68=120+68=188.

188 div by 2,4,47,94, so 0.

x=61: f2(61)=? n - floor(n/8)=61. n=69? 69-8=61, yes. So f2(61)=69.

2*61+69=122+69=191.

191 is prime, so f1(61)=191. increment.

So x=61: f1=191

x=62: f2(62)=? n - floor(n/8)=62. n=70? 70-8=62, yes. So f2(62)=70.

2*62+70=124+70=194.

194 div by 2,97, so 0.

x=63: f2(63)=? n - floor(n/8)=63. n=71? 71-8=63, yes. So f2(63)=71.

2*63+71=126+71=197.

197 is prime, so f1(63)=197. increment.

So x=63: f1=197

x=64: f2(64)=? n - floor(n/8)=64. n=72? 72-9=63? floor(72/8)=9,72-9=63<64. n=73,73-9=64, yes. So f2(64)=73.

2*64+73=128+73=201.

201 div by 3,67, so 0.

x=65: f2(65)=? n - floor(n/8)=65. n=74? 74-9=65, yes. So f2(65)=74.

2*65+74=130+74=204.

204 div by 2,3,4,6,12,17,34,51,68,102, so 0.

x=66: f2(66)=? n - floor(n/8)=66. n=75? 75-9=66, yes. So f2(66)=75.

2*66+75=132+75=207.

207 div by 3,9,23,69, so 0.

x=67: f2(67)=? n - floor(n/8)=67. n=76? 76-9=67, yes. So f2(67)=76.

2*67+76=134+76=210.

210 div by 2,3,5,6,7,10,14,15,21,30,35,42,70,105, so 0.

x=68: f2(68)=? n - floor(n/8)=68. n=77? 77-9=68, yes. So f2(68)=77.

2*68+77=136+77=213.

213 div by 3,71, so 0.

x=69: f2(69)=? n - floor(n/8)=69. n=78? 78-9=69, yes. So f2(69)=78.

2*69+78=138+78=216.

216 div by 2,3,4,6,8,9,12,18,24,27,36,54,72,108, so 0.

x=70: f2(70)=? n - floor(n/8)=70. n=79? 79-9=70, yes. So f2(70)=79.

2*70+79=140+79=219.

219 div by 3,73, so 0.

x=71: f2(71)=? n - floor(n/8)=71. n=80? 80-10=70? floor(80/8)=10,80-10=70<71. n=81,81-10=71, yes. So f2(71)=81.

2*71+81=142+81=223.

223 is prime, so f1(71)=223. increment.

So x=71: f1=223

Now we have a list of x where f1(x) !=0: from above: 1,10,12,14,15,17,19,23,27,29,35,36,42,50,52,57,61,63,71,...

We need to find up to 32nd.

Let's list them in order.

Index of x: let k be the index, so the k-th x where f1(x) !=0.

k=1: x=1

k=2: x=10

k=3: x=12

k=4: x=14

k=5: x=15

k=6: x=17

k=7: x=19

k=8: x=23

k=9: x=27

k=10: x=29

k=11: x=35

k=12: x=36

k=13: x=42

k=14: x=50

k=15: x=52

k=16: x=57

k=17: x=61

k=18: x=63

k=19: x=71

We need more. Continue from x=72.

x=72: f2(72)=? n - floor(n/8)=72. n=82? 82-10=72? floor(82/8)=10.25? floor=10, 82-10=72, yes. So f2(72)=82.

2*72+82=144+82=226.

226 div by 2,113, so 0.

x=73: f2(73)=? n - floor(n/8)=73. n=83? 83-10=73, yes. So f2(73)=83.

2*73+83=146+83=229.

229 is prime, so f1(73)=229. increment.

So k=20: x=73

x=74: f2(74)=? n - floor(n/8)=74. n=84? 84-10=74, yes. So f2(74)=84.

2*74+84=148+84=232.

232 div by 2,4,8,29,58,116, so 0.

x=75: f2(75)=? n - floor(n/8)=75. n=85? 85-10=75, yes. So f2(75)=85.

2*75+85=150+85=235.

235 div by 5,47, so 0.

x=76: f2(76)=? n - floor(n/8)=76. n=86? 86-10=76, yes. So f2(76)=86.

2*76+86=152+86=238.

238 div by 2,7,14,17,34,119, so 0.

x=77: f2(77)=? n - floor(n/8)=77. n=87? 87-10=77, yes. So f2(77)=87.

2*77+87=154+87=241.

241 is prime, so f1(77)=241. increment.

So k=21: x=77

x=78: f2(78)=? n - floor(n/8)=78. n=88? 88-11=77? floor(88/8)=11,88-11=77<78. n=89,89-11=78, yes. So f2(78)=89.

2*78+89=156+89=245.

245 div by 5,7,35,49, so 0.

x=79: f2(79)=? n - floor(n/8)=79. n=90? 90-11=79, yes. So f2(79)=90.

2*79+90=158+90=248.

248 div by 2,4,8,31,62,124, so 0.

x=80: f2(80)=? n - floor(n/8)=80. n=91? 91-11=80, yes. So f2(80)=91.

2*80+91=160+91=251.

251 is prime, so f1(80)=251. increment.

So k=22: x=80

x=81: f2(81)=? n - floor(n/8)=81. n=92? 92-11=81, yes. So f2(81)=92.

2*81+92=162+92=254.

254 div by 2,127, so 0.

x=82: f2(82)=? n - floor(n/8)=82. n=93? 93-11=82, yes. So f2(82)=93.

2*82+93=164+93=257.

257 is prime, so f1(82)=257. increment.

So k=23: x=82

x=83: f2(83)=? n - floor(n/8)=83. n=94? 94-11=83, yes. So f2(83)=94.

2*83+94=166+94=260.

260 div by 2,4,5,10,13,20,26,52,65,130, so 0.

x=84: f2(84)=? n - floor(n/8)=84. n=95? 95-11=84, yes. So f2(84)=95.

2*84+95=168+95=263.

263 is prime, so f1(84)=263. increment.

So k=24: x=84

x=85: f2(85)=? n - floor(n/8)=85. n=96? 96-12=84? floor(96/8)=12,96-12=84<85. n=97,97-12=85, yes. So f2(85)=97.

2*85+97=170+97=267.

267 div by 3,89, so 0.

x=86: f2(86)=? n - floor(n/8)=86. n=98? 98-12=86, yes. So f2(86)=98.

2*86+98=172+98=270.

270 div by 2,3,5,6,9,10,15,18,27,30,45,54,90,135, so 0.

x=87: f2(87)=? n - floor(n/8)=87. n=99? 99-12=87, yes. So f2(87)=99.

2*87+99=174+99=273.

273 div by 3,7,13,21,39,91, so 0.

x=88: f2(88)=? n - floor(n/8)=88. n=100? 100-12=88, yes. So f2(88)=100.

2*88+100=176+100=276.

276 div by 2,3,4,6,12,23,46,69,92,138, so 0.

x=89: f2(89)=? n - floor(n/8)=89. n=101? 101-12=89, yes. So f2(89)=101.

2*89+101=178+101=279.

279 div by 3,9,31,93, so 0.

x=90: f2(90)=? n - floor(n/8)=90. n=102? 102-12=90, yes. So f2(90)=102.

2*90+102=180+102=282.

282 div by 2,3,6,47,94,141, so 0.

x=91: f2(91)=? n - floor(n/8)=91. n=103? 103-12=91, yes. So f2(91)=103.

2*91+103=182+103=285.

285 div by 3,5,15,19,57,95, so 0.

x=92: f2(92)=? n - floor(n/8)=92. n=104? 104-13=91? floor(104/8)=13,104-13=91<92. n=105,105-13=92, yes. So f2(92)=105.

2*92+105=184+105=289.

289 div by 17, so 17 between 2 and 93, so 0.

x=93: f2(93)=? n - floor(n/8)=93. n=106? 106-13=93, yes. So f2(93)=106.

2*93+106=186+106=292.

292 div by 2,4,73,146, so 0.

x=94: f2(94)=? n - floor(n/8)=94. n=107? 107-13=94, yes. So f2(94)=107.

2*94+107=188+107=295.

295 div by 5,59, so 0.

x=95: f2(95)=? n - floor(n/8)=95. n=108? 108-13=95, yes. So f2(95)=108.

2*95+108=190+108=298.

298 div by 2,149, so 0.

x=96: f2(96)=? n - floor(n/8)=96. n=109? 109-13=96, yes. So f2(96)=109.

2*96+109=192+109=301.

301 div by 7,43, so 0.

x=97: f2(97)=? n - floor(n/8)=97. n=110? 110-13=97, yes. So f2(97)=110.

2*97+110=194+110=304.

304 div by 2,4,8,16,19,38,76,152, so 0.

x=98: f2(98)=? n - floor(n/8)=98. n=111? 111-13=98, yes. So f2(98)=111.

2*98+111=196+111=307.

307 is prime, so f1(98)=307. increment.

So k=25: x=98

x=99: f2(99)=? n - floor(n/8)=99. n=112? 112-14=98? floor(112/8)=14,112-14=98<99. n=113,113-14=99, yes. So f2(99)=113.

2*99+113=198+113=311.

311 is prime, so f1(99)=311. increment.

So k=26: x=99

x=100: f2(100)=? n - floor(n/8)=100. n=114? 114-14=100, yes. So f2(100)=114.

2*100+114=200+114=314.

314 div by 2,157, so 0.

This is taking too long. We need to find up to k=32.

Notice that f0(X) is called for X from 0 to 31, and it returns the x value for the (X+1)-th increment, so for X=31, it returns the 32nd x where f1(x) !=0.

From the list, we have up to k=26: x=99

We need k=27,28,29,30,31,32.

Let's try to find pattern or compute faster.

From the condition, f1(x) !=0 when 2x + f2(x) is prime and greater than x+1, but since we check divisors up to x+1, if it is prime, it is not divisible by any d from 2 to x+1 only if the prime is greater than x+1, which it is since 2x + f2(x) > x+1 for x>0.

But from earlier, 2x + f2(x) is approximately 2x + (8/7)x = 22x/7 ≈ 3.14x, so for x>0, it is greater than x+1, so if it is prime, then it is not divisible by any number up to x+1, so f1(x) !=0.

But is it always prime? From above, when it is prime, we have increment, when it is composite, we have 0, but sometimes composite might have no divisors less than x+1, but from our examples, it seems that when it is composite, it usually has a small divisor, but not always? For example, if x is large, and 2x+f2(x) is composite but all divisors are larger than x+1, then it would not be set to 0, but from the check, we check up to x+1, so if all divisors are greater than x+1, then it would not be divisible by any d from 2 to x+1, so f1(x) would be non-zero. But in practice, for composite numbers, there is always a divisor less than or equal to sqrt(n), and if sqrt(n) > x+1, then it might avoid, but since n=2x+f2(x) ≈3.14x, so sqrt(n)≈1.77x, which for x>1, 1.77x > x+1, so it is possible that a composite number has no divisors less than or equal to x+1, so it would not be set to 0.

But from our computation above, for x up to 100, we have only primes so far for non-zero f1, but let's check for larger x.

For example, x=4, 12 composite, has divisors 2,3,4 which are <=5, so set to 0.

x=8,25 composite, has divisor 5<=9, set to 0.

x=9,28 composite, divisors 2,4,7<=10, set to 0.

x=10,31 prime, non-zero.

x=11,34 composite, divisor 2<=12, set to 0.

x=12,37 prime, non-zero.

x=13,40 composite, divisors 2,4,5,8<=14, set to 0.

x=14,43 prime, non-zero.

x=15,47 prime, non-zero.

x=16,50 composite, divisor 2<=17, set to 0.

x=17,53 prime, non-zero.

x=18,56 composite, divisors 2,4,7,8<=19, set to 0.

x=19,59 prime, non-zero.

x=20,62 composite, divisor 2<=21, set to 0.

x=21,65 composite, divisors 5,13<=22, set to 0.

x=22,68 composite, divisor 2<=23, set to 0.

x=23,71 prime, non-zero.

x=24,74 composite, divisor 2<=25, set to 0.

x=25,77 composite, divisors 7,11<=26, set to 0.

x=26,80 composite, divisor 2<=27, set to 0.

x=27,83 prime, non-zero.

x=28,86 composite, divisor 2<=29, set to 0.

x=29,89 prime, non-zero.

x=30,92 composite, divisor 2<=31, set to 0.

x=31,95 composite, divisors 5,19<=32, set to 0.

x=32,100 composite, divisors 2,4,5,10,20,25,50<=33, set to 0.

x=33,103 prime, non-zero.

x=34,106 composite, divisor 2<=35, set to 0.

x=35,109 prime, non-zero.

x=36,113 prime, non-zero.

x=37,116 composite, divisor 2<=38, set to 0.

x=38,119 composite, divisor 7<=39, set to 0.

x=39,122 composite, divisor 2<=40, set to 0.

x=40,125 composite, divisor 5<=41, set to 0.

x=41,128 composite, divisor 2<=42, set to 0.

x=42,131 prime, non-zero.

x=43,135 composite, divisor 3<=44, set to 0.

x=44,138 composite, divisor 2<=45, set to 0.

x=45,141 composite, divisor 3<=46, set to 0.

x=46,144 composite, divisor 2<=47, set to 0.

x=47,147 composite, divisor 3<=48, set to 0.

x=48,150 composite, divisor 2<=49, set to 0.

x=49,153 composite, divisor 3<=50, set to 0.

x=50,157 prime, non-zero.

x=51,160 composite, divisor 2<=52, set to 0.

x=52,163 prime, non-zero.

x=53,166 composite, divisor 2<=54, set to 0.

x=54,169 composite, divisor 13<=55, set to 0.

x=55,172 composite, divisor 2<=56, set to 0.

x=56,175 composite, divisor 5<=57, set to 0.

x=57,179 prime, non-zero.

x=58,182 composite, divisor 2<=59, set to 0.

x=59,185 composite, divisor 5<=60, set to 0.

x=60,188 composite, divisor 2<=61, set to 0.

x=61,191 prime, non-zero.

x=62,194 composite, divisor 2<=63, set to 0.

x=63,197 prime, non-zero.

x=64,201 composite, divisor 3<=65, set to 0.

x=65,204 composite, divisor 2<=66, set to 0.

x=66,207 composite, divisor 3<=67, set to 0.

x=67,210 composite, divisor 2<=68, set to 0.

x=68,213 composite, divisor 3<=69, set to 0.

x=69,216 composite, divisor 2<=70, set to 0.

x=70,219 composite, divisor 3<=71, set to 0.

x=71,223 prime, non-zero.

x=72,226 composite, divisor 2<=73, set to 0.

x=73,229 prime, non-zero.

x=74,232 composite, divisor 2<=75, set to 0.

x=75,235 composite, divisor 5<=76, set to 0.

x=76,238 composite, divisor 2<=77, set to 0.

x=77,241 prime, non-zero.

x=78,245 composite, divisor 5<=79, set to 0.

x=79,248 composite, divisor 2<=80, set to 0.

x=80,251 prime, non-zero.

x=81,254 composite, divisor 2<=82, set to 0.

x=82,257 prime, non-zero.

x=83,260 composite, divisor 2<=84, set to 0.

x=84,263 prime, non-zero.

x=85,267 composite, divisor 3<=86, set to 0.

x=86,270 composite, divisor 2<=87, set to 0.

x=87,273 composite, divisor 3<=88, set to 0.

x=88,276 composite, divisor 2<=89, set to 0.

x=89,279 composite, divisor 3<=90, set to 0.

x=90,282 composite, divisor 2<=91, set to 0.

x=91,285 composite, divisor 3<=92, set to 0.

x=92,289 composite, divisor 17<=93, set to 0.

x=93,292 composite, divisor 2<=94, set to 0.

x=94,295 composite, divisor 5<=95, set to 0.

x=95,298 composite, divisor 2<=96, set to 0.

x=96,301 composite, divisor 7<=97, set to 0.

x=97,304 composite, divisor 2<=98, set to 0.

x=98,307 prime, non-zero.

x=99,311 prime, non-zero.

x=100,314 composite, divisor 2<=101, set to 0.

So from this, it seems that f1(x) !=0 only when 2x+f2(x) is prime. And from the list, we have x where this is prime.

So for f0(X), it returns the (X+1)-th x such that 2x+f2(x) is prime.

But from above, we have the list of x where this is prime.

Now we need the first 32 such x.

From the list:

1,10,12,14,15,17,19,23,27,29,35,36,42,50,52,57,61,63,71,73,77,80,82,84,98,99,...

We have up to x=99 for k=26.

Now continue from x=101 onwards.

x=101: f2(101)=? n - floor(n/8)=101. n=115? 115-14=101? floor(115/8)=14.375 floor=14, 115-14=101, yes. So f2(101)=115.

2*101+115=202+115=317.

317 is prime, so non-zero. So k=27: x=101

x=102: f2(102)=? n - floor(n/8)=102. n=116? 116-14=102, yes. So f2(102)=116.

2*102+116=204+116=320.

320 composite, divisor 2, so 0.

x=103: f2(103)=? n - floor(n/8)=103. n=117? 117-14=103, yes. So f2(103)=117.

2*103+117=206+117=323.

323 div by 17,19, so 0.

x=104: f2(104)=? n - floor(n/8)=104. n=118? 118-14=104, yes. So f2(104)=118.

2*104+118=208+118=326.

326 div by 2, so 0.

x=105: f2(105)=? n - floor(n/8)=105. n=119? 119-14=105, yes. So f2(105)=119.

2*105+119=210+119=329.

329 div by 7,47, so 0.

x=106: f2(106)=? n - floor(n/8)=106. n=120? 120-15=105? floor(120/8)=15,120-15=105<106. n=121,121-15=106, yes. So f2(106)=121.

2*106+121=212+121=333.

333 div by 3,9,37,111, so 0.

x=107: f2(107)=? n - floor(n/8)=107. n=122? 122-15=107, yes. So f2(107)=122.

2*107+122=214+122=336.

336 div by 2,3,4,6,7,8,12,14,16,21,24,28,42,48,56,84,112,168, so 0.

x=108: f2(108)=? n - floor(n/8)=108. n=123? 123-15=108, yes. So f2(108)=123.

2*108+123=216+123=339.

339 div by 3,113, so 0.

x=109: f2(109)=? n - floor(n/8)=109. n=124? 124-15=109, yes. So f2(109)=124.

2*109+124=218+124=342.

342 div by 2,3,6,9,18,19,38,57,114,171, so 0.

x=110: f2(110)=? n - floor(n/8)=110. n=125? 125-15=110, yes. So f2(110)=125.

2*110+125=220+125=345.

345 div by 3,5,15,23,69,115, so 0.

x=111: f2(111)=? n - floor(n/8)=111. n=126? 126-15=111, yes. So f2(111)=126.

2*111+126=222+126=348.

348 div by 2,3,4,6,12,29,58,87,116,174, so 0.

x=112: f2(112)=? n - floor(n/8)=112. n=127? 127-15=112, yes. So f2(112)=127.

2*112+127=224+127=351.

351 div by 3,9,13,27,39,117, so 0.

x=113: f2(113)=? n - floor(n/8)=113. n=128? 128-16=112? floor(128/8)=16,128-16=112<113. n=129,129-16=113, yes. So f2(113)=129.

2*113+129=226+129=355.

355 div by 5,71, so 0.

x=114: f2(114)=? n - floor(n/8)=114. n=130? 130-16=114, yes. So f2(114)=130.

2*114+130=228+130=358.

358 div by 2,179, so 0.

x=115: f2(115)=? n - floor(n/8)=115. n=131? 131-16=115, yes. So f2(115)=131.

2*115+131=230+131=361.

361 div by 19, so 0.

x=116: f2(116)=? n - floor(n/8)=116. n=132? 132-16=116, yes. So f2(116)=132.

2*116+132=232+132=364.

364 div by 2,4,7,13,14,26,28,52,91,182, so 0.

x=117: f2(117)=? n - floor(n/8)=117. n=133? 133-16=117, yes. So f2(117)=133.

2*117+133=234+133=367.

367 is prime, so non-zero. So k=28: x=117

x=118: f2(118)=? n - floor(n/8)=118. n=134? 134-16=118, yes. So f2(118)=134.

2*118+134=236+134=370.

370 div by 2,5,10,37,74,185, so 0.

x=119: f2(119)=? n - floor(n/8)=119. n=135? 135-16=119, yes. So f2(119)=135.

2*119+135=238+135=373.

373 is prime, so non-zero. So k=29: x=119

x=120: f2(120)=? n - floor(n/8)=120. n=136? 136-17=119? floor(136/8)=17,136-17=119<120. n=137,137-17=120, yes. So f2(120)=137.

2*120+137=240+137=377.

377 div by 13,29, so 0.

x=121: f2(121)=? n - floor(n/8)=121. n=138? 138-17=121, yes. So f2(121)=138.

2*121+138=242+138=380.

380 div by 2,4,5,10,19,20,38,76,95,190, so 0.

x=122: f2(122)=? n - floor(n/8)=122. n=139? 139-17=122, yes. So f2(122)=139.

2*122+139=244+139=383.

383 is prime, so non-zero. So k=30: x=122

x=123: f2(123)=? n - floor(n/8)=123. n=140? 140-17=123, yes. So f2(123)=140.

2*123+140=246+140=386.

386 div by 2,193, so 0.

x=124: f2(124)=? n - floor(n/8)=124. n=141? 141-17=124, yes. So f2(124)=141.

2*124+141=248+141=389.

389 is prime, so non-zero. So k=31: x=124

x=125: f2(125)=? n - floor(n/8)=125. n=142? 142-17=125, yes. So f2(125)=142.

2*125+142=250+142=392.

392 div by 2,4,7,8,14,28,49,56,98,196, so 0.

x=126: f2(126)=? n - floor(n/8)=126. n=143? 143-17=126, yes. So f2(126)=143.

2*126+143=252+143=395.

395 div by 5,79, so 0.

x=127: f2(127)=? n - floor(n/8)=127. n=144? 144-18=126? floor(144/8)=18,144-18=126<127. n=145,145-18=127, yes. So f2(127)=145.

2*127+145=254+145=399.

399 div by 3,7,19,21,57,133, so 0.

x=128: f2(128)=? n - floor(n/8)=128. n=146? 146-18=128, yes. So f2(128)=146.

2*128+146=256+146=402.

402 div by 2,3,6,67,134,201, so 0.

x=129: f2(129)=? n - floor(n/8)=129. n=147? 147-18=129, yes. So f2(129)=147.

2*129+147=258+147=405.

405 div by 3,5,9,15,27,45,81,135, so 0.

x=130: f2(130)=? n - floor(n/8)=130. n=148? 148-18=130, yes. So f2(130)=148.

2*130+148=260+148=408.

408 div by 2,3,4,6,8,12,17,24,34,51,68,102,136,204, so 0.

x=131: f2(131)=? n - floor(n/8)=131. n=149? 149-18=131, yes. So f2(131)=149.

2*131+149=262+149=411.

411 div by 3,137, so 0.

x=132: f2(132)=? n - floor(n/8)=132. n=150? 150-18=132, yes. So f2(132)=150.

2*132+150=264+150=414.

414 div by 2,3,6,9,18,23,46,69,138,207, so 0.

x=133: f2(133)=? n - floor(n/8)=133. n=151? 151-18=133, yes. So f2(133)=151.

2*133+151=266+151=417.

417 div by 3,139, so 0.

x=134: f2(134)=? n - floor(n/8)=134. n=152? 152-19=133? floor(152/8)=19,152-19=133<134. n=153,153-19=134, yes. So f2(134)=153.

2*134+153=268+153=421.

421 is prime, so non-zero. So k=32: x=134

So the 32nd x where f1(x) !=0 is x=134.

Therefore, for f0(31), it returns the 32nd such x, which is 134.

But f0(X) for X=31 returns x-1? Let's see the code of f0.

def f0(X):
x,i = 0,0
while i <= max(X,0):
if (1 - f1(x)) <= 0:
i = i + 1
x = x + 1
return x - 1

So for X=31, i <=31, so when i becomes 32, loop stops, and we return x-1.

And i is incremented when f1(x) >=1, so when we have the 32nd such x, i becomes 32, then x is incremented, then loop stops, return x-1.

So if the 32nd x is 134, then when x=134, f1(134)>=1, so i is incremented to 32, then x is set to 135, then we check while condition: i=32 and X=31, so i>31, so loop stops, return x-1=135-1=134.

So f0(31)=134.

Similarly, for other X, f0(X) returns the (X+1)-th x where f1(x) !=0.

So for the output, we need f0(x) for x from 0 to 31.

So from above, we have the list of x where f1(x) !=0, and f0(X) returns the (X+1)-th one.

So for X=0, f0(0)= first x=1

X=1, f0(1)= second x=10

X=2, f0(2)= third x=12

X=3, f0(3)= fourth x=14

X=4, f0(4)= fifth x=15

X=5, f0(5)= sixth x=17

X=6, f0(6)= seventh x=19

X=7, f0(7)= eighth x=23

X=8, f0(8)= ninth x=27

X=9, f0(9)= tenth x=29

X=10, f0(10)= eleventh x=35

X=11, f0(11)= twelfth x=36

X=12, f0(12)= thirteenth x=42

X=13, f0(13)= fourteenth x=50

X=14, f0(14)= fifteenth x=52

X=15, f0(15)= sixteenth x=57

X=16, f0(16)= seventeenth x=61

X=17, f0(17)= eighteenth x=63

X=18, f0(18)= nineteenth x=71

X=19, f0(19)= twentieth x=73

X=20, f0(20)= twenty-first x=77

X=21, f0(21)= twenty-second x=80

X=22, f0(22)= twenty-third x=82

X=23, f0(23)= twenty-fourth x=84

X=24, f0(24)= twenty-fifth x=98

X=25, f0(25)= twenty-sixth x=99

X=26, f0(26)= twenty-seventh x=101

X=27, f0(27)= twenty-eighth x=117

X=28, f0(28)= twenty-ninth x=119

X=29, f0(29)= thirtieth x=122

X=30, f0(30)= thirty-first x=124

X=31, f0(31)= thirty-second x=134

So for the output, we print these values for x from 0 to 31.

So the output should be:

f0(0)=1

f0(1)=10

f0(2)=12

f0(3)=14

f0(4)=15

f0(5)=17

f0(6)=19

f0(7)=23

f0(8)=27

f0(9)=29

f0(10)=35

f0(11)=36

f0(12)=42

f0(13)=50

f0(14)=52

f0(15)=57

f0(16)=61

f0(17)=63

f0(18)=71

f0(19)=73

f0(20)=77

f0(21)=80

f0(22)=82

f0(23)=84

f0(24)=98

f0(25)=99

f0(26)=101

f0(27)=117

f0(28)=119

f0(29)=122

f0(30)=124

f0(31)=134

So when we run the code, it will print these values for x from 0 to 31.

So the algorithm is computing the sequence of numbers where2x + f2(x) is prime, and then returning the (n+1)-th such number for input n.

But from the code, it prints f0(x) for x from 0 to 31.

So the output is as above.