Lcm of all numbers so far

Definition

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

• The least common multiple of all numbers from $1$ to $n$, i.e., as:

$\mathrm{l}\mathrm{c}\mathrm{m}\{1,2,\dots ,n\}$.

• The product of all primes powers $p^k$ for which $p^k\le n$ but $p^{k+1}>n$.
• The exponent of the symmetric group of degree $n$.
• 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 $n=0,1,2,3,4,5,6,7$ are $1,1,2,6,12,60,60,420$.

Growth

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

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

• Factorial
• Primorial, Primorial of the second kind

Logarithm

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