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The sequence is infinite since for every element k with '0' as last digit or no '0' at all every k*10^m is also in it.
Table of n, a(n) for n=1..35.
3114 is okay, since it has three different digits while 3114^2 = 9696996 has only two.
(PARI) isok(n) = length(Set(digits(n, 10))) > length(Set(digits(n^2, 10))) \\ Michel Marcus, Jul 22 2013
Sequence in context: A141948 A102908 A141998 * A142945 A142079 A142224
Adjacent sequences: A061274 A061275 A061276 * A061278 A061279 A061280
Ulrich Schimke (ulrschimke(AT)aol.com)