Mersenne numbers and the (divisors) property.

[franktaw at netscape.net]

Gerald McGarvey <Gerald.McGarvey at comcast.net> wrote:
>Something to note about A007733:
(Period of 1/n in base 2).
>For 2 <= n <= 84, A000010(n) / A007733(n) results in the following sequence:
>...it seems reasonable to conjecture that A007733 always divides A000010 
>(Euler totient function phi(n)).

This is true.  In slightly more generality:

The period of 1/n in base b is the (multiplicative order of b' modulo n, where b' is b with all factors of n factored out.  (E.g., b = 12, n = 10, factor all 2's and 5's out of b gives b' = 3.)

Since b' is always a unit modulo n, it's order divides the order of the group, which is phi(n).

Q.E.D.

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