Pentagon Number Theorem

Pentagon Number Theorem
Transferred from Wikipedia http://zh.wikipedia.org/wiki/pentagram Theorem

Pentagon Number TheoremIt is a mathematical theorem discovered by Euler and describes the features of Euler's function expansion.^{[1][2]. The expansion of the Euler's function is as follows:}

That is

After the Euler's function is expanded, some power items are removed, leaving only 1, 2, 5, 7, 12 ,... the power left behind is exactly the generalized Pentagon number.

If the formula above is regarded as a power series, its convergence radius is 1. However, if it is considered as a form idempotent series, its convergence radius is not considered.

Relationship with the split Function

The reciprocal of the Euler's function is the primary function of the split function, that is:

It is a K-separated function.

The formula above matches the Pentagon Number Theorem to obtain

When N> 0, the coefficient on the right of the equation is 0. Compare the coefficient on the two sides of the equation.

Therefore, we can obtain the recursive formula of the split function p (n ).

Take n = 10 as an Example

In this way, the P (n) value can be obtained in log (n) time.