The Formula
Let n ≥ 1 be an integer. The prime decomposition of n! is given by
where
and the brackets represent the floor function. Note that the former product can equally well be taken only over primes less than or equal to n, and the latter sum can equally well be taken for j ranging from 1 to logp(n), i.e :
Note that, for any real number x, and any integer n, we have:
which allows one to more easily compute the terms sp(n).
The small disadvantage of the De Polignac's formula is that we need to know all the primes up to n. In fact,
where is a prime-counting function counting the number of prime numbers less than or equal to n
Read more about this topic: De Polignac's Formula
Famous quotes containing the word formula:
“I cannot give you the formula for success, but I can give you the formula for failure—which is: Try to please everybody.”
—Herbert B. Swope (1882–1958)
“Ideals possess the strange quality that if they were completely realized they would turn into nonsense. One could easily follow a commandment such as “Thou shalt not kill” to the point of dying of starvation; and I might establish the formula that for the proper functioning of the mesh of our ideals, as in the case of a strainer, the holes are just as important as the mesh.”
—Robert Musil (1880–1942)