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September 8th, 2018, 08:46 AM  #1 
Newbie Joined: Sep 2018 From: US Posts: 1 Thanks: 0  Sum of a nondecreasing sequence of integers
For which pairs of positive integers $(n, d)$ is it true that every integer $S$ can be written as a sum of a nondecreasing sequence of integers $a_1+a_2+...+a_n$ and $a_na_1=d$?

September 8th, 2018, 03:09 PM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,112 Thanks: 1002 
That's sure confusing, Jimbo. Are you talking about an arithmetic sequence? Say sequence is 1,2,3 a1=1, a2=2, an = 3 (n = 3) 1 + 2 + 3 = 6 = S ? (n,d) = 3,2)? What does "every integer S" mean? Any integer? 
September 8th, 2018, 05:56 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2606 Math Focus: Mainly analysis and algebra 
I have a vague feeling that there may be no such pairs. I have a stronger feeling that I'm probably wrong about that. 
September 8th, 2018, 06:06 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,112 Thanks: 1002  
September 8th, 2018, 07:09 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,616 Thanks: 2606 Math Focus: Mainly analysis and algebra 
Nothing stronger than tea tonight. (Sorry Jonah, you can have mine).


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integers, nondecreasing, sequence, sum 
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