Interesting series 42, 4422, 444222....

On observing keenly, I found that the number series 42, 4422, 444222, $\cdots, \cdots$ is  interesting. I dont know if any number theorists have realized this fact or not. Because these are the only numbers, I suppose that are the product of consecutive two numbers and the first half of the number of digits are exactly double the second half of the number of digits.

$6  \times 7            = 42$ ;          $\Rightarrow\frac{4}{2}=2$ .

$66 \times 67        =4422$;         $\Rightarrow\frac{44}{22}=2$.

$666 \times 667     =444222$;     $\Rightarrow\frac{444}{222}=2$.

$6666 \times 6667 =44442222$; $\Rightarrow\frac{4444}{2222}=2$.

and this holds true for any number of digits. The number of digits in product is equal to the sum of the number of multipliers and the multiplicands. This is very interesting fact.

If there are some series that show the similar characteristics, please do not hesitate to comment.

Along with this, it is also observed that 67 has a very interesting property when it is multiplies with the multiples of 3 as observed in the following sequence
$67 \times 3     =  0201 ;\\              67 \times 6     =  0402 ; \\ 67 \times 9     =  0603 ; \\  67 \times 12   =  0804 ; \\  67 \times 15   =  1005; \\  67 \times 18   =  1206; \\  67 \times 21   =  1407; \\  67 \times 24   =  1608; \\  67 \times 27   =  1809; \\  67 \times 30   =  2010; \\  67 \times 33   =  2211; \\  67 \times 36   =  2412; \\  67 \times 39   =  2613; \\  67 \times 42   =  2814; \\ \cdots \\ \cdots \\ \cdots \\ \mathbf{67} \times \mathbf{66}=\mathbf{4422}; \\ \cdots \\  67 \times 99   =  6633; \\  67 \times 102 =  6834 ; \\ \cdots \\ \cdots \\ \cdots \\  67 \times 297 =19899$.

   In each of the cases the first two digits are excatly twice the last two digits and this continues for the multiples of 3. The reason for this is the number 201;

This series continues till $67 \times 297 =19899$ or till $201 \times 99=19899$; After this it does not show the similar nature.

Similar other interesting pattern observed is:

$21         =      3\times  7 \\ 2211     =      33 \times 67 \\ 222111 =     333 \times 667 \\ 22221111 = 3333 \times 6667 \\ \cdots\\ \cdots\\ \cdots\\ $

Also

$63         =      9 \times 7 \\ 6633     =      99  \times 67 \\ 666333 =     999 \times 667 \\ 66663333 = 9999 \times 6667 \\ \cdots \\ \cdots \\ \cdots$

This series is also interesting

$84         =      12  \times 7 \\ 8844     =     \underline{132} \times 67   (132=1(2+1)2 \; \text{or} \; 1212 \;  \text{with the middle two taking carry from previous} 1) \\ 888444 =     1332 \times 667 \\ 88884444 =  13332  \times 6667 \\ \cdots \cdots \cdots$

and so on

Similar interesting pattern can be observed till

$189         =    27  \times 7  \\ 19899    =     297  \times 67  (198 \; \text{is observed from 1818 where 1 is carried over to 8 }\\ \hspace{3.2cm} \text{and similarly 297 is obtained from 2727 where 2 is carried over to 7 }) \\ 1998999 =    2997  \times 667 \\ 19998999 =  29997 \times 6667 \\ \cdots\\ \cdots\\ \cdots$