Using the definitions at Wikipedia; Sloane; and Mathworld; I can't see why $1$ is a member of the Kaprekar series?
Would someone give an easy explanation?
Thanks.
(Yet more on this here).
$1=0+1$, $1^2=0\times10^m+1$ seems to fit the definition as given at the OEIS reference.
Unfortunately, my answer is anticlimactic. I happen to know wikipedia's definition (linked in the question, but I reproduce the definition here)
Let X be a non-negative integer. X is a Kaprekar number for base b if there exist non-negative integers n, A, and positive number B satisfying: $X^2 = Ab^n + B$, where $0 < B < b^n$, and s.t. $X = A + B$
So $A$ can be $0$. Thus $1^2 = 1 = 0* 10^1 + 1$, and we see that it's a Kaprekar number.
And - Gerry posted his answer just before me (I refreshed, and it's there)! But I wrote this too, so I'll keep it -