How does "The partition numbers" relate to prime numbers?
There is a mathematical relationship between the prime numbers and the partitions of integers. The partition number of an integer n, written p(n), counts the number of ways you can divide the positive integers less than or equal to n into non-empty subsets such that no two elements in the same subset are equal.

For example, consider the partition number of 8: there are 3 possible partitions of 8 into 2 parts (8 = 2+2): {2}, {4}, and {6}. Similarly, the partition number of 13 is 6, because it can be divided evenly into 3 parts: {3}, {5}, and {7}.

The connection between the prime numbers and the partition numbers comes from the fact that every prime number can be expressed as a product of smaller primes, and the largest power of a prime that divides a given integer is equal to its partition number. For example, 25 can be expressed as 5*25, since 5 is a factor of 25 and 25 has three different partitions {5}, {10}, and {15} corresponding to its divisors 5, 10, and 15. On the other hand, 17 cannot be expressed as a product of smaller primes, so its partition number must be larger than any of the smaller powers of 2 that divide it. Thus, p(17) = 6.

So, while the prime numbers and the partition numbers are separate concepts with their own properties and connections, they do have some similarities in how they're related to each other.