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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).

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    I don't see what an Italian auto manufacturer has to do with it. $1$ is included because it satisfies the definitions, as both answerers agree.2011-06-08

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$1=0+1$, $1^2=0\times10^m+1$ seems to fit the definition as given at the OEIS reference.

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    Umm, your right. But I do like the Iannucci paper just stating it is 'cos it is :-)2011-06-08
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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 -

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    Yep. I now see where I was wrong. Thanks.2011-06-08