Crazy Librarian's Interesting Prime Permutation Index Number Generator

Question

You saved the day with your prime sequence code, and the math teacher loved it. So much so that a new challenge was posed to the librarian (a/k/a, your boss). Congratulations, you get to code the solution so the librarian can once again impress the math teacher.

Start with the sequence of natural numbers in base-10, \$\mathbb N\$

0, 1, 2, 3, 4, 5, 6 ...

Excluding 0 and 1, every number in this sequence either is prime, \$\mathbb P\$

2, 3, 5, 7, 11, 13 ...

or composite, \$\mathbf C\$

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 ...

Reflecting on how the librarian thought to insert an integer digit into the decimal expansion of a number from \$\mathbb P\$, the math teacher instead created a function \$G(x,y)\$ that takes a number \$x\$ from \$\mathbb N\$ with \$1 \le x \le 9\$ and a number \$y\$ from \$\mathbf C\$ and inserts \$x\$ into the decimal expansion of \$y\$ in every position, in order left-to-right, selecting only unique numbers.

For example, \$G(3,14)\$ is \$314, 134, 143\$. However, \$G(1,14)\$ is only \$114, 141\$, as whether you prepend or insert the \$1\$ into \$14\$, the same number \$114\$ is generated.

The math teacher wondered how many times you'd have to do these permutations before you get a number that's in \$\mathbb P\$, if you took \$x\$ in increasing order. The math teacher called this the Composite-Prime Index of a number, and wrote it as \$\text{CPI}(y)\$.

For example, \$4\$ only needs to be done twice: \$14, 41\$, since \$41\$ is prime, so \$\text{CPI}(4)\$ is \$2\$. However, \$8\$ needs to be done 6 times, \$18, 81, 28, 82, 38, 83\$ before reaching \$83\$ as a prime number, so \$\text{CPI}(8)\$ is \$6\$.

Your task is to write code that will output this Composite-Prime Index, given an input number.

Input

• A single integer \$y\$, such that \$y\$ is in \$\mathbf C\$, input via function argument, STDIN, or equivalent.
• For the purposes of calculation, you can assume \$y\$ will fit in usual integer ranges (e.g., assume \$2^{31}-1\$ as an upper bound).
• Behavior for \$y\$ not in C is undefined.

Output

The resultant Composite-Prime Index, calculated as described above, output to STDOUT or equivalent, with two exceptions:

• If the very last permutation (i.e., appending \$9\$ to \$y\$) is the one that results in a prime, output -1. An example, expanded below, is \$y=14\$.
• If there is no permutation (i.e., \$G(x,y)\$ is a subset of \$\mathbf C\$ for all \$1 \le x \le 9\$), output 0. An example, expanded below, is \$y=20\$.

Examples

 y -> operations             : output
 4 -> 14, 41                 : 2
 6 -> 16, 61                 : 2
 8 -> 18, 81, 28, 82, 38, 83 : 6
 9 -> 19                     : 1
10 -> 110, 101               : 2
12 -> 112, 121, 212, 122, 312, 132, 123, 412, 142, 124, 512, 152, 125, 612, 162, 126, 712, 172, 127 : 19
14 -> 114, 141, 214, 124, 142, 314, 134, 143, 414, 144, 514, 154, 145, 614, 164, 146, 714, 174, 147, 814, 184, 148, 914, 194, 149 : -1
15 -> 115, 151               : 2
16 -> 116, 161, 216, 126, 162, 316, 136, 163 : 8
18 -> 118, 181               : 2
20 -> 120, 210, 201, 220, 202, 320, 230, 203, 420, 240, 204, 520, 250, 205, 620, 260, 206, 720, 270, 207, 820, 280, 208, 920, 290, 209 : 0

Restrictions

• This is code-golf, since you'll need to transcribe this to an index card so the librarian can show the math teacher, and your hand cramps easily.
• Standard loophole restrictions apply. The librarian doesn't tolerate cheaters.

Leaderboard

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Answer 1

Haskell, 166 161 bytes

p n=mod(product[1..n-1]^2)n>0
q=p.read
n#c=[h++c:t|i<-[0..length n],(h,t)<-[splitAt i n]]
[y]%i|q y= -1|1<2=0
(y:z)%i|q y=i|1<2=z%(i+1)
f n=((n#)=<<['1'..'9'])%1 

Usage examples: f "8" -> 6, f "14"-> -1, f "20"-> 0.

How it works: p is the primality test (stolen from @Mauris' answer in a different challenge). q a wrapper for p to convert types from strings to integer. n # c inserts c at every position in n. % takes a list of numbers and an index i. When the first element of the list is prime, return i, else recure with the tail of the list and i+1. Stop when there's a single element left and return -1 if it's prime and 0 otherwise.

Answer 2

Pyth, 35 bytes

&sKm}dPdsm{msj`dcz]khlzS9|%hxK1lK_1

Test suite

Answer 3

Minkolang 0.11, 85 bytes

n1(l*$d`)d9[i3G(0c2c$%$r2c*l*2c3c1+*++2gl:d2G)2gx1c2G3gx]r3XS(2M4&I)N.ikI1-4&1~N.1+N.

Try it here.

Explanation (coming soon)

n            Take integer from input (say, n)
1(           Calculate smallest power of 10 greater than n (say, a)
  l*         Multiply by 10
    $d`      Duplicate stack and push n>a
       )     Close while loop (ends when n<=a)
        d    Duplicates a (let's call it b)

9[                                                 For loop that runs 9 times 
  i1+                                              Loop counter + 1 (say, i)
     3G                                            Puts the loop counter in position 3
       (                                           Opens while loop
        0c2c$%                                     Copies n and b and pushes n//b, n%b
              $r                                   Swaps top two elements of stack
                2c*l*                              Copies b and multiplies by 10
                     2c3c*                         Copies b and i and multiplies them
                          ++                       Adds it all together (inserts i)
                            2gl:                   Gets b and divides by 10
                                d2G                Duplicates and puts one copy back
                                   )               Closes while loop (breaks when b=0)
                                    2gx            Gets and dumps b
                                       1c2G        Copies a and puts it in b's place
                                           3gx     Get and dumps i
                                              ]    Close for loop

r       Reverses stack
 3X     Dumps the top three elements (namely, n, a, and b)
   S    Removes duplicates

(                           Opens while loop
 2M                         Pushes 1 if top of stack is prime, 0 otherwise
   4&                       Jump four spaces if prime
     I)N.                   If the loop actually finishes, then all were composite,
                             so output 0 and stop.
         ik                 Pushes loop counter and breaks
           I1-              Pushes length of stack minus 1 (0 if last one was prime)
              4&1~N.        If this is 0, pushes -1, outputs as integer, and stops.
                    1+N.    Adds 1, outputs as integer, and stops.

Answer 4

Javascript, 324 bytes

y=>(p=(n,c)=>n%c!=0?c>=n-1?1:p(n,++c):0,u=a=>a.filter((c,i)=>a.indexOf(c)==i),g=(x,y)=>u(((x,y,z)=>z.map((c,i)=>z.slice(0,i).join("")+x+z.slice(i).join("")).concat(y+x))(x,y,y.split(''))),h=(x,y)=>g(x,y).concat(x==9?[]:h(++x,y)),i=h(1,y).reduce((r,c,i)=>r?r:p(c,2)?i+1:0,0),console.log(p(y,2)||y<2?'':i==h(1,y).length?-1:i))

If y not in C, then STDOUT output is empty.

Explanation

y=>(
    //Prime Test function
    p=(n,c)=>n%c!=0?c>=n-1?1:p(n,++c):0,

    //Unique function
    u=a=>a.filter((c,i)=>a.indexOf(c)==i),

    //Generates numbers from a couple x and y
    g=(x,y)=>u(((x,y,z)=>z.map((c,i)=>z.slice(0,i).join("")+x+z.slice(i).join("")).concat(y+x))(x,y,y.split(''))),

    //Generates all possible numbers from y using recusion
    h=(x,y)=>g(x,y).concat(x==9?[]:h(++x,y)),

    //Check if any prime in the generated numbers
    i=h(1,y).reduce((r,c,i)=>r?r:p(c,2)?i+1:0,0),

    console.log(
        //Is Y in C ?
        p(y,2)||y<2?
            ''
            :
            // Check if the answer is not the last one
            i==h(1,y).length?-1:i)
    )

Answer 5

Jelly, 25 bytes

D-ṭJœṖ€$jþ9ẎṖ€ḌQẒµL,-yi1$

Try it online!

+6 bytes to allow for:

If the very last permutation (i.e., appending 9 to y) is the one that results in a prime, output -1

How it works

D-ṭJœṖ€$jþ9ẎṖ€ḌQẒµL,-yi1$ - Main link. Takes n on the left
D                         - Digits of n
 -ṭ                       - Append -1; Call this D
       $                  - To D:
   J                      -   Indices; Yield [1, 2, ..., len(D)]
      €                   -   For each index i:
    œṖ                    -     Partition D at index i
          9               - Generate the range [1,2,3,4,5,6,7,8,9]
         þ                - Pair up all pairs from the range and the partitions:
        j                 -   Join each partition with the corresponding integer
           Ẏ              - Tighten into a list of lists
            Ṗ€            - Remove the -1 from each
              Ḍ           - Convert back to integers
               Q          - Deduplicate
                Ẓ         - Replace primes with 1 and composites with 0
                 µ        - Call this list of bits B
                  L       - Length of B
                   ,-     - Pair with -1
                        $ - To B:
                      i1  -   Yield the first index of 1, or 0 if not found
                     y    - If the index is equal to the length, replace with -1