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Partition generation

Let the partition function P(n)P(n) enumerate the ways nn can be expressed as a distinct sum of positive integers, e.g. P(4)=5P(4) = 5 since 4=3+1=2+2=2+1+1=1+1+1+14 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 are the only ways to represent 44.

p prime[n=0P(n)pn]\prod_{p \ \text{prime}} \left[ \sum_{n=0}^{\infty} P(n)p^{-n} \right]

Does the above product converge?

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