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I recently ran into an interesting type of number that people from where I was born like to refer to as números cabalísticos. They are supposedly "magic" kinds of numbers that possess mystical characteristics that are tightly bound to the celestial or spiritual realms (I could go on and on about it). However, I have only encountered one such number, and it is this one:

$142857$

Notice that if you multiply this number by any of the integers from $2$ to $6$, you will get a different permutation of it, and if you multiply it by the number $7$ (which is a “divine” number), you will get an interesting result: $142857\cdot2=285714$ $142857\cdot3=428571$ $142857\cdot4=571428$ $142857\cdot5=714285$ $142857\cdot6=857142$ $142857\cdot7=999999$ My questions are: is there a “theory” behind this kind of numbers? Could they be helpful in any way? Are there any others? Is the world ending this year? Thanks, guys!

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    See http://en.wikipedia.org/wiki/Full_reptend_prime and http://en.wikipedia.org/wiki/Cyclic_number2012-03-05

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The theory is in your last line, $142857 \cdot 7 = 999999$, which means that $\frac 17 = 0.\overline{142857}$ Any prime which has full period will do the same. The next is $\frac {1}{17}=0.\overline{0588235294117647}$ which you can multiply by any number from $2$ through $16$ and get a cyclic shift.

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    But now you've taken the *cabalístico* out of it...2015-07-21
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Not only 142857 but there are many more similar and even an unproven conjecture is that there are infinitely many. In mathematics these numbers are known as cyclic numbers and there are many references to them in particular see here