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Sum of k numbers has exactly k divisors (Posted on 2024-03-03) Difficulty: 3 of 5
Do there exist positive integers A1, A2,....., A100 such that for each k= 1,2,....,100, the number A1+A2+....+Ak has precisely k divisors?
Provide sufficient reason for your assertion.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution No problem (spoiler) Comment 1 of 1
Yes.  many such sequences.

The simplest and possibly smallest is 1,1,2,4,8,16,32 ...

Then the partial sums are 1,2,4,8,16, etc, and the kth partial sum has k divisors.

We are done.

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However, one can construct any sequence of partial sums Sk such that Sk has k divisors and Sk > Sk-1.  
Then  Ak = Sk - Sk-1  works




  Posted by Steve Herman on 2024-03-03 07:38:04
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