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Added: 1a2,5

:I doubt that there's a known formula.
*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A023814 for the total number of semigroups on a set with n elements
*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027851 for the number of non-isomorphic semigroups with n elements --AxelBoldt

Question: How many semigroups are there of a given finite order? Is there a formula?

: I doubt that there's a known formula.

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A023814 for the total number of semigroups on a set with n elements
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027851 for the number of non-isomorphic semigroups with n elements --AxelBoldt