Timeline for Density of set of unique sums
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 6, 2018 at 12:52 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
added 1 character in body
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| Jun 3, 2016 at 6:37 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
deleted 42 characters in body
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| Jun 2, 2016 at 13:30 | comment | added | Ilya Bogdanov | After a correction: If my computations are (now) correct, we have $A_2=\{3,4\}$, $A_3=\{7,8\}$, $A_4=\{12,14,15,16\}$, $A_5=\{13,21,26,27,29,31,32\}$, $A_6=\{50,54,55,56,57,59,60,61,62,63,64\}$. | |
| Jun 2, 2016 at 13:28 | comment | added | Ilya Bogdanov | @MartinSleziak: Oh, sorry! 28 also wasn't in my $A_5$. it was just a typo. So hopefully $A_6$ is still correct. | |
| Jun 2, 2016 at 13:21 | comment | added | Martin Sleziak | @IlyaBogdanov I think that $28\notin A_5$, since $28=14+14=12+16$. Otherwise I got the same sets $A_2$ to $A_5$. I did not try $A_6$. | |
| Jun 2, 2016 at 12:56 | comment | added | Ilya Bogdanov | Several issues worth being mentioned: a) appearance of $13$ in $A_5$, although $\max U_4=16>13$ ($13$ is not representable by $U_3$), as well as of $51$ in $A_7$; b) NOT appearance of $25$ in $U_6$ (it was not representable by $U_4$, but it is multiply representable by $U_5$); and c) NOT appearance of $49$ in $U+U$.(no numbers below $49$ can appear in $A_n$ with $n\geq 7$). This may warn from too simple approaches. | |
| Jun 2, 2016 at 11:47 | comment | added | Martin Sleziak | If you read the comments to my answer, you can see that writing out the set $U_n$ at least for a few small $n$'s could help to clarify your question. (Specifically Emil Jeřábek asked whether when you write "sum of two elements", these two elements are supposed to be distinct. In other words, whether $4=2+2$ belongs to $A_2$ or not.) | |
| Jun 2, 2016 at 10:44 | comment | added | Martin Sleziak | It turns out (see Emil Jeřábek's comment) that what I posted is not an answer to your question. | |
| Jun 2, 2016 at 10:12 | vote | accept | Dominic van der Zypen | ||
| Jun 2, 2016 at 11:37 | |||||
| Jun 2, 2016 at 6:41 | comment | added | Martin Sleziak | Just to check whether I understood the problem correctly, are these first values correct? $U_1=\{1,2\}$, $U_2=\{3,4\}$, $U_3=\{6,7,8\}$, $U_4=\{12,13,15,16\}$, $U_5=\{24,25,26,27,29,30,31,32\}$, $U_6=\{48,49,63,64\}$ | |
| Jun 2, 2016 at 6:39 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
added conjecture
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| Jun 2, 2016 at 6:37 | comment | added | Dominic van der Zypen | Thanks Martin for your comment; I conjecture the value is $0$ -- I edited the post accordingly. | |
| Jun 2, 2016 at 6:33 | comment | added | Martin Sleziak | Is there some conjecture about the value? Did you try some experiments (by hand or using the computer) to calculate $|U_n|$ at least for some small $n$'s? (The next logical step would be checking whether that sequence appears in OEIS.) | |
| Jun 1, 2016 at 12:42 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |