As a start, search for Periodic Table Island of Stability.
http://en.wikipedia.org/wiki/Island_of_stability
references some discussions. “In nuclear physics, the island of stability is a set of as-yet undiscovered heavier isotopes of transuranium elements which are theorized to be much more stable than some of those closer in atomic number to uranium. Specifically, they are expected to have radioactive decay half-lives of minutes or days, with “some optimists” expecting half-lives of millions of years.”
“The possibility of an “island of stability” was first proposed by Glenn T. Seaborg in the late 1960s.”
“… One possible magic number of neutrons for spherical nuclei is 184, and some possible matching proton numbers are 114, 120 and 126 – which would mean that the most stable spherical isotopes would be flerovium-298, unbinilium-304 and unbihexium-310. Of particular note is Ubh-310, which would be “doubly magic” (both its proton number of 126 and neutron number of 184 are thought to be magic) and thus the most likely to have a very long half-life. (The next lighter doubly magic spherical nucleus is lead-208, the heaviest known stable nucleus and most stable heavy metal.)”
“Recently, interest has been expressed over the possibility of a second island of stability. The idea of a second island of stability was presented by Yuri Oganessian at the 235th national meeting of the American Chemical Society. This new island would be centered around element 164 (Unhexquadium), especially isotope Uhq-482, with a stability similar to that of flerovium.”
Here’s the scary part:
“ In particular, the very small critical masses of transplutonic elements (possibly as small as grams) implies that if stable elements could be found, they would enable small and compact nuclear bombs either directly or by serving as primaries to help ignite fission/fusion secondaries; this possibility motivated much of the early research and multiple nuclear tests by the United States (including Operation Plowshare) and the Soviet Union aimed at producing such elements.”
Imagine a tiny nuclear bomb requiring only a few grams of one of these elements! Currently, something like a pound and a half of fissionable material is required. See what I mean, if we create elements like these (for peaceful uses) then we can create far more powerful spacecraft to explore space.
I suppose a half life with millions of years instead of nanoseconds might be possible. This “unexpectanium”, however, would likely be as extremely dangerous as difficult to produce.