Fungrim entry: 7932c3

g(n) = \max \left\{ \operatorname{lcm}\!\left({s}_{1}, \ldots, {s}_{k}\right) : k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {s}_{i} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \sum_{i=1}^{k} {s}_{i} = n \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

g(n) = \max \left\{ \operatorname{lcm}\!\left({s}_{1}, \ldots, {s}_{k}\right) : k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; {s}_{i} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \sum_{i=1}^{k} {s}_{i} = n \right\}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LandauG`	$g(n)$	Landau's function
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Sum`	$\sum_{n} f(n)$	Sum

Source code for this entry:

Entry(ID("7932c3"),
    Formula(Equal(LandauG(n), Maximum(Set(LCM(Subscript(s, 1), Ellipsis, Subscript(s, k)), For(Tuple(k, Subscript(s, i))), And(Element(k, ZZGreaterEqual(0)), Element(Subscript(s, i), ZZGreaterEqual(1)), Equal(Sum(Subscript(s, i), For(i, 1, k)), n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

Landau's function