142,857


142857, the six repeating digits of, 0., is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of,,,, or respectively.
142,857 is a Kaprekar number and a Harshad number.

Calculation

If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857:
Multiplying by a multiple of 7 will result in 999999 through this process:
If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.
It is the repeating part in the decimal expansion of the rational number = 0.. Thus, multiples of are simply repeated copies of the corresponding multiples of 142857:

as an infinite sum

There is an interesting pattern of doubling, shifting and addition that gives .
Each term is double the prior term shifted two places to the right. This is can be proved by applying the identity for the sum of a geometric sequence:
Another infinite sum is

Other bases

In some other bases, six-digit numbers with similar properties exist, given by. For example, in base 12 it is 186A35 and base 24 3A6LDH.

Connection to the enneagram

The 142857 number sequence is used in the enneagram figure, a symbol of the Gurdjieff Work used to explain and visualize the dynamics of the interaction between the two great laws of the Universe, the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by,. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.