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?

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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

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