> Let phi(m,k) = 
> the number of positive integers j, where j <= m,
> and GCD(j,k) = 1.
...
> For a given positive integer m, how many positive integers, k, are
> there where:
> phi(m,k) = k ?
> I THINK the sequence begins (figured by hand):
> 1, 0, 1, 2, 0, 1, 1, 2, 0, 0,....

It continues 0 1 1 1 1 1 1 2 0 1 0 0 1 2 1 1 0 1 0 1 3 3 0 0 1 1 1 1...

I'm submitting it as A077614 now.

I don't see any simple pattern. The maximums seem to increase slowly.
The least inverse begins:

1 4 31 255 431 432 1727 2700 21599 21600 10800 10799 43200 43199 ...

(going in as a077615)

The zeroes never seem to run out.