The [https://mathworld.wolfram.com/PartitionFunctionP.html Partition Function P] is the function P(n), where n∈ℤ, defined as the number of distinct ways in which n can be expressed as the sum of non-increasing positive integers. 


;Example:
  P(4) = 5   because   4 = <big>Σ</big>(4) = <big>Σ</big>(3,1) = <big>Σ</big>(2,2) = <big>Σ</big>(2,1,1) = <big>Σ</big>(1,1,1,1)


P(n) can be expressed as the recurrence relation: 
  P(n) = P(n-1) +P(n-2) -P(n-5) -P(n-7) +P(n-12) +P(n-15) -P(n-22) -P(n-26) +P(n-35) +P(n-40) ...

The successive numbers in the above equation have the differences: &nbsp; 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8 ...

This task may be of popular interest because [https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg Mathologer] made the video, [https://www.youtube.com/watch?v=iJ8pnCO0nTY The hardest "What comes next?" (Euler's pentagonal formula)], where he asks the programmers among his viewers to calculate P(666). The video was viewed more than 100,000 times in the first couple of weeks after its release.

In Wolfram Language, this function has been implemented as PartitionsP.


;Task:
Write a function which returns the value of PartitionsP(n). Solutions can be iterative or recursive. 

Bonus task: show how long it takes to compute PartitionsP(6666).


;References:
* [https://www.youtube.com/watch?v=iJ8pnCO0nTY The hardest "What comes next?" (Euler's pentagonal formula)] The explanatory video by Mathologer that makes this task a popular interest.
* [https://mathworld.wolfram.com/PartitionFunctionP.html Partition Function P] Mathworld entry for the Partition function.
* [[wp:Partition_function_(number_theory)|Partition function (number theory)]] Wikipedia entry for the Partition function.


;Related tasks:
* [[9 billion names of God the integer]]

<br><br>