Here
http://wwwhomes.uni-bielefeld.de/achim/highly.html
there is a quite shocking link that states that a false algorithm was published and even a number theoretic expert failed to detect that.
In view of this , how reliable is the external link of the $10,000$ first highly composite numbers here
https://en.wikipedia.org/wiki/Highly_composite_number
and the subsequent external link even containing over $700,000$ HCN. It is also claimed that the calculation is extremely fast.
I asked a question about an efficient algorithm for checking whether a number is highly composite and the answer was that it is very difficult to check this.
But if the used algorithm actually guarantees to find the first over $700,000$ HCN-numbers, the algorithm used should be very efficient considering the calculation times.
Can anyone explain the main idea of the algorithm used in the calculation of the huge list of highly composite numbers ? In particaular, how can I prove the given estimation for the exponents due to Ramanujan ?
I also invite everyone to comment on the "joke" that was apparantly made.
make an answer. Most anything you might want will be among the publications of Jean-Louis Nicolas
He published a summary in English called On Highly Composite Numbers which is currently number 80 on his publication page. His student, Guy Robin, is the one who constructed an algorithm for finding all HC numbers between consecutive SHC numbers. I believe Robin later wrote up that algorithm, which was originally in his dissertation.
I don't know what algorithms might be involved in your links. I should emphasize that, done correctly and carefully, finding the SHC numbers is fast enough; given that you have found one, the next one is what you have multiplied by a single prime. If I had to find all HC numbers up to a large bound and it needed to be correct, I would get someone to translate Robin's note for me. I already know how to find the SHC numbers up to some bound, and have done so.