Timeline for About some functions in the set of the natural numbers
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 22, 2012 at 17:34 | vote | accept | Dimiter Skordev | ||
| Nov 22, 2012 at 17:34 | |||||
| Nov 21, 2012 at 20:13 | comment | added | Emil Jeřábek | Let $\alpha=\liminf_nf(n)/n$, $\beta=\limsup_nf(n)/n$. Clearly $1\le\alpha\le\beta\le2$, and I believe your argument gives $\beta\le1+\beta/(1+\alpha)$ and $\alpha\ge1+\alpha/(1+\beta)$. This implies $1+\beta\le\alpha\beta\le1+\alpha$, thus $\alpha=\beta=\phi$, i.e., $f(n)\sim\phi n$. | |
| Nov 21, 2012 at 18:54 | vote | accept | Dimiter Skordev | ||
| Nov 22, 2012 at 16:31 | |||||
| Nov 21, 2012 at 15:17 | comment | added | Andreas Blass | The argument I gave for the growth rate, approximately linear with the golden ratio as coefficient, was done under the assumption of Presburger definability, an assumption that I showed is false. Nevertheless, the growth rate looks plausible even without the false assumption, and its plausibility is certainly increased by Emil's comment (which arrived while I was typing my answer). | |
| Nov 21, 2012 at 15:12 | history | answered | Andreas Blass | CC BY-SA 3.0 |