There is a long history of identifying subsets of the natural numbers particularly based on their divisibility properties. There are friendly numbers, extravagant numbers, deficient, perfect... The other day in class we were doing prime factorizations and I wound up with something like 2^2*3^3. I started thinking about all numbers n such that if n has a prime factor p, p^p divides n. Even more, I was thinking about adding the stipulation of sequential prime factors. So here are the first 4 numbers...

2^2=4

2^2*3^3=108

2^2*3^3*5^5=337,500

2^2*3^3*5^5*7^7=277,945,762,500

Anyone ever heard of numbers like this? What are they called? Are they used for anything?

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## 2 comments:

The closest I found in the Online Encyclopedia of Integer Sequences, here, was the hyperfactorials, which also show up on wikipedia

Great find. So these numbers are the hyperfactorials with a prime domain. Thanks

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