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