# Lcm of all numbers so far

From Number

## Contents

## Definition

Let be a natural number. The **lcm of all numbers so far** for 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 ID of the sequence in the Online Encyclopedia of Integer Sequences is A003418

### Initial values

The values for are .

### 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.