# Lcm of all numbers so far

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## Definition

Let  be a natural number. The lcm of all numbers so far for  to , i.e., as:

.

## 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.