Timeline for Decomposition of a natural number as sum of positive integers
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2023 at 18:42 | comment | added | Sylvain JULIEN | I may have mixed up admissible $k$-tuples and prime constellations. I wrote my comment in my train after work and somewhat lacked focus. | |
| Mar 1, 2023 at 17:46 | comment | added | Steven Clark | @SylvainJULIEN Both of the lists I posted in the comments above contain numbers that are not in OEIS entry A008407 (see oeis.org/A008407). For example, both lists contain the number $4$. Or did I misinterpret your comment? | |
| Mar 1, 2023 at 17:09 | comment | added | Sylvain JULIEN | @StevenClark the values you listed look like diameters of prime constellations. Is it merely coincidental? | |
| Feb 28, 2023 at 14:03 | answer | added | Pavel Gubkin | timeline score: 1 | |
| Feb 24, 2023 at 7:37 | answer | added | Pavel Gubkin | timeline score: 7 | |
| Feb 24, 2023 at 5:33 | comment | added | Joachim König | I suggest to additionally allow $a, b, c, d$ to be $0$, and show that with this convention, $f(n) =\varphi(n)$ for all $n>1$ (equivalently, when disallowing $0$, $f(n) =\varphi(n)-2$ for all $n>2$ as observed by Peter Taylor) . Indeed, there seems to be a 1-1 correspondence between such $(a, b, c, d)$ and coprime residues mod $n$ via $(a, b, c, d) \mapsto a+b$. Surely this can be seen via some simple trick? | |
| Feb 23, 2023 at 21:55 | comment | added | Peter Taylor | @StevenClark, for $n > 2$ that simplifies to $f(n) \stackrel{?}{=} \varphi(n) - 2$, which holds for $3 \le n \le 4500$. | |
| Feb 23, 2023 at 17:28 | comment | added | Steven Clark | @PeterTaylor Thanks, with this correction for the first 30 values I get $\{0,0,0,0,2,0,4,2,4,2,8,2,10,4,6,6,14,4,16,6,10,8,20,6,18,10,16,10,26,6\}$ which seems to correspond to $f(n)=\text{A181830}(n)+\text{A070824}(n-1)$ except for $n=1$ (see oeis.org/A070824 ). | |
| Feb 23, 2023 at 17:27 | comment | added | Pavel Gubkin | @FFCH, the question would be much nicer if you could add the first values of $f(n)$ into the question | |
| Feb 23, 2023 at 17:06 | comment | added | Peter Taylor |
@StevenClark, no. All of the < should be <=.
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| Feb 23, 2023 at 16:35 | comment | added | Martin Rubey | isn't it simply en.wikipedia.org/wiki/B%C3%A9zout%27s_identity? | |
| Feb 23, 2023 at 16:31 | comment | added | Steven Clark | Here's my Mathematica code for verification: $$f(\text{n$\_$})\text{:=}\text{Block}[\{a=1,b,c,d,\text{cnt}=0\},\text{While}[a<n-3,b=1;\text{While}[a+b<n-2,c=1;\text{While}[a+b+c<n-1,d=n-(a+b+c);\text{If}[a d-b c=1,\text{cnt}\text{++}];c\text{++}];b\text{++}];a\text{++}];\text{cnt}]$$ Does it correctly implement the function $f(n)$? | |
| Feb 23, 2023 at 16:22 | comment | added | Steven Clark | @MartinRubey For the first 30 values I get $\{0,0,0,0,1,0,2,2,2,1,6,2,6,4,4,4,11,4,12,6,6,6,18,6,12,9,14,8,22,6\}$ which seems to correspond to oeis.org/A181830 except for $n=1$. | |
| Feb 23, 2023 at 16:12 | comment | added | Martin Rubey | oeis.org/A055684 | |
| Feb 23, 2023 at 15:41 | comment | added | Puzzled | Did you consider the condition $ad-bc=1$? | |
| Feb 23, 2023 at 15:35 | comment | added | Peter Taylor | If my calculations are correct and the sequence starts [2, 0, 4, 2, 4, 2, 8, 2, 10, 4, 6, 6, 14, 4, 16, 6, 10, 8, 20, 6, 18, 10, 16, 10, 26, 6, 28, 14, 18, 14, 22, 10, 34, 16, 22, 14, 38, 10, 40, 18, 22, 20, 44, 14, 40, 18, 30, 22, 50, 16, 38, 22, 34, 26, 56] with offset 5 then it's not in OEIS. | |
| Feb 23, 2023 at 15:34 | review | Close votes | |||
| Feb 28, 2023 at 3:07 | |||||
| Feb 23, 2023 at 15:19 | comment | added | Puzzled | I want to take the order into account. So the partitions of $10$ you wrote are different. I added a condition on $a,b,c,d$ to my question. | |
| Feb 23, 2023 at 15:18 | history | edited | Puzzled | CC BY-SA 4.0 |
added 22 characters in body
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| Feb 23, 2023 at 15:11 | comment | added | Stefan Kohl♦ | Are you counting ordered or unordered partitions, i.e. do you count 10 = 1+2+3+4 and 10 = 4+3+2+1 as the same or as two distinct decompositions? | |
| Feb 23, 2023 at 15:05 | history | asked | Puzzled | CC BY-SA 4.0 |