Display all prime numbers

Context

A prime number is a number that allows only 2 distinct divisors: 1 and itself.
First prime numbers are 2, 3, 5, 7, 11 ...
Computing all prime numbers up to a limit can be implemented using a sieve, that works like this

• Generate an array of all integers from 2 up to the limit
• Go to the first non-null item in the array
• This number N is prime
• Then, reset array items every N cells starting from this one

Example: 2 is prime, and all multiple of 2 are removed, then 3 is a prime and all multiples of 3 are removed, including 6 that will be reset twice (once for 2, and once for 3), then 4 is null (reset by 2), so move to 5, ...

Initial state

• Memory: empty
• Cursor: first cell
• Input: any

Process

• First, define a 'comma' on first cell (code: 44), because we'll display comma-separated numbers, so it avoids building the comma everytime.
• Note: this is equivalent to a constant value in other programming languages...
• Then, generate all numbers from limit to 2, in decreasing order, in a 2-cells array, but this time it'll be [{value, 1}]  instead of [{1, value}] (code is a bit shorter)
• Generate the limit (using '-', we'll have 255, the biggest possible value)
• Set next cell to 1
• While current value is not null
• Copy 2 cells ahead
• Set next cell to 1
• Decrease copied value by 1, and stay on this value
• Result will be an array from 255 to 0, let's just remove the extra cells at the end.
• Finally, go to the first cell
• If value is not null
• Print the number
• Print comma
• Copy / move it twice: one for backup, one for counter
• Iterate on all array cells
• Move 2-cells block to the right
• Decrease counter. If null
• Reset copied cell
• Reset counter to initial value (copy from backup)
• Move whole array at its original position
• Go to next cell

Code - try it

generate comma
>++++[-<+++++++++++>]
generate all numbers from max to 2
>-[[->+>+<<]>[-<+>]+>-]<-<-<

[
  go to current number
  -<
  [
    number is not null: prime
    copy twice and print
    [->>+>>+<<<<]
    >>>>[>>>>++++++++++<<<<[->+>>+>-[<-]<[->>+<<<<[->>>+<<<]>]<<]>+[-<+>]>>>[-]>[-<<<<+>>>>]<<<<]<[>++++++[<++++++++>-]<-.[-]<]
    print comma
    <<<<[<<]<.>>>[>>]
    copy counter
    >[->+>>+<<<]>[-<+>]<

    for each cell
    <<<[
      move
      -<[->>+<<]>>>+
      go to counter and decrease then check if null
      [>>]+<-[>-]>[-
        reset last copied cell
        <<[<<]>[-]>[>>]
        reset counter
        >>[-<+<<+>>>]<[->+<]
      ]
      process next cell
      <<<[<<]<<
    ]
    move back all cells
    >>>>[<[-<<+>>]<+>>->>]
    clear counters
    <[-]>>>[-]<<<<<
  ]
  process next number
  <
]

Final state

• Memory: 44 0 0 0 (44 is code for comma)
• Cursor: second cell
• Input: unchanged
• Output: all primes below 255, comma separated, with extra comma - note that extra comma can be removed by adding a Backspace character (code = 10). Simply add ++++++++++. at the end of the code.