Question:
If p(n) is the number of partitions of n, defined as the number of ways to write the integer n as a sum of positive integers where the order of the addends is not significant, what n produces the 42nd largest prime p(n)?
Answer:
The 42nd largest prime p(n) is 3513035269942590955686749126214187667970579050845937, which is produced by n = 2508.
The number of partitions p(n), as stated in the question, is defined as the number of ways to write the integer n as a sum of positive integers where the order of the addends is not significant. So, for example, for n = 4, p(4) = 5 as illustrated below:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
A prime number is a positive integer that has no positive divisors other than 1 and itself. p(n) is prime for certain values of n, the first few of which are 2, 3, 4, 5, 6, 13, 36, 77, 111… corresponding to p(n) equal to 2, 3, 5, 7, 11, 101, 17977, 10619863,...
You can follow the latest top 20 list of largest p(n) primes at http://primes.utm.edu/top20/page.php?id=54.
