Hi!

I try to find the integer factorization for all this numbers.

So i found :

1-1
12-2^2x3^1
123-3^1x41^1
1234-2^1x617^1
12345-3^1x5^1x823^1
123456-2^6x3^1x643^1
1234567-127^1x9721^1
12345678-2^1x3^2x47^1x14593^1
123456789-3^2x3607^1x3803^1

1-1
21-3^1x7^1
321-3^1x107^1
4321-29^1x149^1
54321-3^1x19^1x953^1
654321-3^1x218107^1
7654321-19^1x402859^1
87654321-3^2x1997^1x4877^1
987654321-3^2x17^2x379721^1

After this i read about powerfull numbers here

http://en.wikipedia.org/wiki/Powerful_number

So from all this numbers, only 1 is a powerfull number because in the integer factorization af all others number there is at least one factor with exponent 1.

And from all this number, only 1 is a perfect square!

So I belive, the wanted number is 1.