This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just give brute force code which (at least briefly playing with PARI) isn't going to scale up beyond $10^{10}$ or so on sane timescales. This suggests there's some ways of doing it more efficiently (any links or tips?)... or is this what supercomputers are for?

up vote 1 down vote accepted

You can use the fact that the sum of divisors function is multiplicative. We have that $\sigma(p^n)=\frac {p^{n+1}-1}{p-1}$ for $p$ prime and $\sigma(rs)=\sigma(r)\sigma(s)$ for $r$ coprime to $s$. So you look for combinations of prime powers that cancel off the denominators. You need lots of prime factors, which therefore need to be pretty small, so you can ignore a lot of numbers.

  • Indeed. Having now solved a Project Euler problem involving such numbers, it's quite surprising how small the contributing factors remain. – timday Apr 27 '13 at 18:05

Some answers can be found in Ron Sorli's thesis at http://hdl.handle.net/2100/275.

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.