Timeline for Recovering n from sigma(n)/n
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 4, 2013 at 4:23 | comment | added | Jose Arnaldo Bebita | Hi @TomDeMedts. You might want to refer to Kurt Ludwick's "Analysis of the ratio $\frac{\sigma(n)}{n}$" for an algorithm. | |
| Feb 24, 2011 at 10:01 | comment | added | François Brunault | @Tom : yes, looking for such $n$ is quite restrictive. The next step would be to search for integers of the form $n=b'c$ with $b'$ divisible by $b$ but not containing other prime factors than those of $b$, and with $(b',c)=1$. You can set a reasonable bound on the exponents of the factorization of $b'$, say 5 or 6. However I never wrote such a program myself, so I don't know if this is really efficient. | |
| Feb 24, 2011 at 8:42 | vote | accept | Tom De Medts | ||
| Feb 24, 2011 at 8:42 | comment | added | Tom De Medts | By the way, the "$b$" in my previous comment is really the "$b$" from the comment below my question, so $q=a/b$ with $a,b$ coprime. So only looking at $n=bc$ with $b,c$ coprime really is a restriction. | |
| Feb 23, 2011 at 20:31 | comment | added | Tom De Medts | @Francois: What I'm doing at the moment is a combination of recursively writing $n = bc$ with $b,c$ coprime (maybe the coprimeness is too restrictive, but it makes the computation easier), together with a fast lookup-table containing the values for $\delta(n)$ for $n$ going from $1$ to $10^7$. But I find this somehow rather down-to-the-earth, and I was wondering whether there exist more advanced methods instead. | |
| Feb 23, 2011 at 18:22 | comment | added | François Brunault | @Tom : A natural idea to compute integers $n$ such that $\delta(n)=q$ is to find (recursively) conditions on the decomposition of $n$ into prime factors (this is just an extension of the last remark in your question). I think such a procedure is very familiar to people searching for perfect and multiperfect numbers (which I'm not, so there may be improvements on this idea). | |
| Feb 23, 2011 at 15:51 | comment | added | Tom De Medts | Thanks a lot! Are there more recent developments since Pomerance's paper (which is from 1977)? Do you know of any kind of efficient algorithms to try to compute, for example, the smallest element in $\delta^{-1}(q)$ for given $q$ (up to some given upper bound)? | |
| Feb 23, 2011 at 13:43 | comment | added | Sidney Raffer | Pomerance's website math.dartmouth.edu/~carlp contains a direct link to this paper and 175 (!) others. | |
| Feb 23, 2011 at 11:49 | history | answered | Gjergji Zaimi | CC BY-SA 2.5 |