Timeline for Asympotic density of a very simple sequence
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 19, 2019 at 21:10 | vote | accept | Yaakov Baruch | ||
| Mar 19, 2019 at 20:47 | answer | added | Stanley Yao Xiao | timeline score: 2 | |
| S Mar 19, 2019 at 20:14 | history | suggested | Ali Taghavi |
I add two tags.
|
|
| Mar 19, 2019 at 19:45 | review | Suggested edits | |||
| S Mar 19, 2019 at 20:14 | |||||
| Mar 17, 2019 at 12:56 | comment | added | Yaakov Baruch | Substituting $N^{1/3}x$ for $n$: $S\approx \frac{N^{2/3}}{2}\int_{N^{-1/3}}^{N^{1/6}}\big(\sqrt{x^2+\frac{4}{x}}-x\big)dx\approx \frac{N^{2/3}}{2}\int_0^\infty\big(\sqrt{x^2+\frac{4}{x}}-x\big)dx\approx 2.6499581...\times N^{2/3}$. Empirically, if $T$ is the count without multiplicities, $T/(2.6499581...\times N^{2/3})$ seems to be increasing to a limit $\le1$, so that likely $T=O(N^{2/3})$. The definite integral above seems to match OEIS sequence A197374. | |
| Mar 17, 2019 at 12:56 | comment | added | Yaakov Baruch | Expanding on Lucia's insightful comment, $mn(m+n)\le N$ implies both $n\lessapprox\sqrt{N}$ and $m\le \frac{1}{2}\big(-n+\sqrt{n^2+\frac{4N}{n}}\big)$, therefore if $S$ is the count with multiplicities of the number of terms in $a$ up to $N$, then $S\approx \frac{1}{2}\int_1^{\sqrt{N}}\big(\sqrt{n^2+\frac{4N}{n}}-n\big)dn$... (cont) | |
| Mar 17, 2019 at 8:06 | comment | added | Yaakov Baruch | @AaronMeyerowitz: a sorted set, so no multiplicies. | |
| Mar 17, 2019 at 1:54 | comment | added | Aaron Meyerowitz | I notice that $30$ is only listed once on spite of 2,3 and 1,5. So are you taking multiplicity into account or not? | |
| Mar 16, 2019 at 21:48 | comment | added | Lucia | I'd guess that the multiplicity with which an integer can be written as $mn(m+n)$ is usually small (maybe just $2$ typically). In any case the multiplicity is on average no more than powers of $\log$. Counting with multiplicity it would be easy to get an asymptotic for the number of $m$ and $n$ with $mn(m+n) \le x$: this is readily seen to be a constant times $x^{2/3}$. This is in keeping with what you notice on $a(n)$. Same comments apply to $AA$. | |
| Mar 16, 2019 at 20:58 | history | edited | Yaakov Baruch | CC BY-SA 4.0 |
deleted 34 characters in body
|
| Mar 16, 2019 at 20:45 | history | asked | Yaakov Baruch | CC BY-SA 4.0 |