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:
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)
““It’s hard enough to adjust [to the lack of control] in the beginning,” says a corporate vice president and single mother. “But then you realize that everything keeps changing, so you never regain control. I was just learning to take care of the belly-button stump, when it fell off. I had just learned to make formula really efficiently, when Sarah stopped using it.””
—Anne C. Weisberg (20th century)