Lcm of all numbers so far

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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.