We can do this strange concatenation without focus on the prime factors. One can define left and right concatenation and create a table.
\[\begin{array}{|c|cc|} \hline &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & [0] & [1] & 2 & 3 & [4]& 5 & [6]& 7 & [8]& [9] \\ 1 & [10] & 11 & [12] & 13 & [14] & [15] & [16] & 17 & [18] & 19 \\ 2 & [20] & [21] & [22] & 23 & [24] & [25] & [26] & [27] & [28] & 29 \\ 3 & [30] & 31 & [32] & [33] & [34] & [35] & [36] & 37 & [38] & [39] \\ 4 & [40] & 41 & [42] & 43 & [44] & [45] & [46] & 47 & [48] & [49] \\ 5 & [50] & [51] & [52] & 53 & [54] & [55] & [56] & [57] & [58] & 59 \\ 6 & [60] & 61 & [62] & [63] & [64] & [65] & [66] & 67 & [68] & [69] \\ 7 & [70] & 71 & [72] & 73 & [74] & [75] & [76] & [77] & [78] & 79 \\ 8 & [80] & [81] & [82] & 83 & [84] & [85] & [86] & [87] & [88] & 89 \\ 9 & [90] & [91] & [92] & [93] & [94] & [95] & [96] & 97 & [98] & [99] \\ \hline \end{array}\]
Of course this is the definition of a list/grid. But such a device is the basis for some seive like techniques. A grid like this makes me think of the sequence,
2|0,0,1,0,1,0,3,0,1,0,3,0,1,0,3,0,5,0,1,0,5,0,3,0,1,0,3,0,5,0,5,0,1,0,5,0,3,0,1,0,5,0,3,0,5,0,7,0,...
Which should be the number of consecutive compounds between each prime. where a zero is the prime, thus we can remove the zeroes...
We obtain, 0,1,1,3,1,3,1,3,5,1,5,3,1,3,5,5,1,5,3,1,5,3,5,7...
which is OEIS:A046933.
However let us turn the above matrix into a prime or not prime map \[\begin{array}{|c|cccccccccc|c||cc|} \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & bin & n_2& HEX\\ \hline 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0110^* & 6 & 6\\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1111 & 15 & F\\ 2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0101 & 5 & 5\\ 3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1010 & 10 & A\\ 4 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1110 & 14 & E\\ 5 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0101 & 5 & 5\\ 6 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1010 & 10 & A\\ 7 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1101 & 13 & D\\ 8 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0101 & 5 & 5\\ 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0010 & 2 & 2\\ \hline \end{array}\]
We can picture this matrix as the top slice of a cubic array, this slice had an address of zero prepending the number entries, the next will have and address of 1, thus what was \(00\) will now be \(100\), what was \(89\) will now be \(189\) etc. We can attempt to see how the primality map shifts down the layers.
\[\begin{array}{|c|cccccccccc|c||cc|} \hline 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & bin & n_2 & HEX\\ \hline 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1111 & 15 & F\\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0100 & 4 & 4\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0010 & 2 & 2\\ 3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1011 & 11 & B\\ 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0001 & 1 & 1\\ 5 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1010 & 10 & A\\ 6 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0110 & 6 & 6\\ 7 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0101 & 5 & 5\\ 8 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1000 & 8 & 8\\ 9 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1111 & 15 & F\\ \hline \end{array}\]
The sequence of up to the next \(4\) primes can be stored in a single hex digit past the first row, as primes can only end in \(1,3,7,9\) from that point onwards. This gives the hexadecimal sequence: 6,F,5,A,E,5,A,D,5,2,F,4,2,B,1,A,6,5,8,F... this appears to be OEIS: A105052