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The periodic table as currently used is based on the chemistry of the different elements. The effect of the number of neutrons on the chemistry of the elements is almost completely negilgible (actually, for rather subtle reasons relating to the quantisation of bond energies this is not quite true for very light elements, particularly hydrogen, but near enough). Re-classifying the periodic table according to some sort of scheme that included the number of neutrons as well as the number of protons would mean that carbon-12 (6 protons, 6 neutrons) would appear as a seperate "element" from carbon-13 (6 protons, 7 neutrons).
Which would be of no use or interest to chemists (who are the people who use the periodic table),
and whilst such a classification might be of interest to nuclear physicists, I don't think they're much interested, for the most part, in the chemistry of the elements they're working with (except once they get to the point of trying to engineer an application). So no, I don't think this is an idea with any legs to speak of.
The physicists already have such a table, and I wish them the best of luck memorising it.http://en.wikipedia.org/wiki/Table_of_nuclides_(complete)
A single neutron is not stable and decays by the emission of an electron to become a proton with a half life of about 14 minutes. Because it has no orbiting electron it does not take part in normal chemical reactions and therefore has no place in a normal periodic table of stable elements with chemical properties. It does take part in nuclear reactions and has its place in a periodic table of nuclear types. These have very different groupings of stability these shells and structural elements are partially known see http://en.wikipedia.org/wiki/Nuclear_structure for more information.