In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn.
Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence.
No other partition of 5 yields a bigger lcm, so g(5) = 6.
An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5).
There are arbitrarily long sequences of consecutive numbers n, n + 1, ..., n + m on which the function g is constant.
[1] 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, ... (sequence A000793 in the OEIS) is named after Edmund Landau, who proved in 1902[2] that (where ln denotes the natural logarithm).
denotes the prime counting function,
the logarithmic integral function with inverse
for some constant c > 0 by Ford,[4] then[3] The statement that for all sufficiently large n is equivalent to the Riemann hypothesis.