perplexus.info :: Numbers : Sum of k numbers has exactly k divisors

Sum of k numbers has exactly k divisors (Posted on 2024-03-03)

Do there exist positive integers A₁, A₂,....., A₁₀₀ such that for each k= 1,2,....,100, the number A₁+A₂+....+A_k has precisely k divisors?
Provide sufficient reason for your assertion.

No Solution Yet Submitted by K Sengupta

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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 S_{k such that}S_k_{has k divisors and}S_{k >}S_k-1.

_ThenA_{k =}S_{k -}S_{k-1 works}

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

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