# Lcm of all numbers so far

From Number

## Definition

Let be a natural number. The **lcm of all numbers so far** for is defined as:

- The least common multiple of all numbers from to , i.e., as:

.

- The product of all primes powers for which but .
- The exponent of the symmetric group of degree .
- It is the exponential of the second Chebyshev function.

## Behavior

The initial values of the lcm of all numbers so far for are given as: 1, 1, 2, 6, 12, 60, 60, 420, [SHOW MORE] View list on OEIS### Growth

The lcm of all numbers so far has approximately exponential growth in . Moreover, it is not strictly increasing as a function of , and it increases in value *only* at prime powers. At the prime power , it gets multiplied by .

It is the exponential of the second Chebyshev function. More details on the growth are to be found in the page on the second Chebyshev function.

## Relation with other functions

### Logarithm

The logarithm of the lcm of all numbers so far is equal to the second Chebyshev function.