- Landau's function
Landau's function "g"("n") is defined for every
natural number "n" to be the largest order of an element of thesymmetric group "S""n". Equivalently, "g"("n") is the largestleast common multiple of any partition of "n".For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so "g"(5) = 6. An element of order 6 in the group "S"5 can be written in cycle notation as (1 2) (3 4 5).
The
integer sequence "g"(0) = 1, "g"(1) = 1, "g"(2) = 2, "g"(3) = 3, "g"(4) = 4, "g"(5) = 6, "g"(6) = 6, "g"(7) = 12, "g"(8) = 15, ... is [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793 A000793] .The sequence is named after
Edmund Landau , who proved in 1902 (reference [1] below) that:(where ln denotes thenatural logarithm ).The statement that :
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