Prime permutation - Random walk

[y.kohmoto]

    Hello, seqfans.

    I considered a permutation of primes.
    I think that a random walk makes an array  of primes, so it becomes a
permutation.



    %S A000001 3, 7, 19, 43,
    %N A000001
    If Prime(n-1)= +1 , Mod 4, and is arranged on k-th line, then we put
Prime(n) on (k-1)-th line.
    If Prime(n-1)= -1 , Mod 4, and is arranged on k-th line, then we put
Prime(n) on (k+1)-th line.

    First term Prime(2)=3 is arranged on 0-th line.

     %C A000001
    This process makes an array of prime numbers.

    3, 7, 19, 43,                                          ....0-th line
    5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127,           ....1st line
    13, 29, 37, 53, 61, 71, 79, 101, 107, 113               ....2nd line
    73, 83, 97, 109,                                       ....3rd line
    89,                                                    ....4-th line

    The sequence gives numbers on 0-th line.
    Probably it is a kind of random walk, isn't it?

    I think the sequence is a mapping from Prime sequence to {x,y : 0<x}
plane.
    So, if we count the primes like the following sequence, it will become a
permutation of primes.



    s(9)                               -3rd
    s(4) s(10)                         -2nd
    s(1) s(5) s(11)                    -1th
    s(0) s(2) s(6) s(12)               0-th
    s(3) s(7) s(13)                    1st
    s(8) s(14)                         2nd
    s(15)                              3rd

    Numbers are rather small, so no primes appears on -1st line or -2nd
line.
    I want to know how the permutation is.

     %S A000002 5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127,

     Sequence gives numbers on 1st line.

         --------------------

    %S A000003 5, 11, 17, 23, 41, 47, 83,
    If Prime(n-1)= +1 , Mod 3, and is arranged on k-th line, then we put
Prime(n) on (k-1)-th line.
    If Prime(n-1)= -1 , Mod 3, and is arranged on k-th line, then we put
Prime(n) on (k+1)-th line.

    First term Prime(3)=5 is arranged on 0-th line.
    Sequence gives numbers on 0-th line.

    5, 11, 17, 23, 41, 47, 83,
    7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101,
    31, 59, 67, 73, 97,
    61,

    %S A000004 7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101,

    Sequence gives numbres on 1st line.

         -------------------

    If Prime(n-1)=  1 or 2 , Mod 5, and is arranged on k-th line, then we
put Prime(n) on (k-1)-th line.
    If Prime(n-1)=  -1 or -2  , Mod 5, and is arranged on k-th line, then we
put Prime(n) on (k+1)-th line.

    First term Prime(4)=7 is arranged on 0-th line.

    31, 97,
    7, 29, 37, 61, 89, 101,                                 ....0-th line
    11, 17, 23, 41, 47, 59, 67, 83,
    13, 19, 43, 53, 71, 79,
    73,

    Neil :
    Do they fit to OEIS?

    Yasutoshi