How to find $\sum_{k=1}^{N} \left \lfloor \frac{N}{k} \right \rfloor * k \mid k $ is composite

Asked Apr 12 '20 at 02:20

Active Apr 12 '20 at 02:41

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I have to find the following value

$\sum_{k=1}^{N} \left \lfloor \frac{N}{k} \right \rfloor * k \mid k $ is composite and all powers of prime factor of k are equal and greater than 1

I tried to find a closed form which seems impossible, so is it possible to compute it in subroot complexity.

For example: $36 = 2^{2} * 3^{2}$

edited Apr 12 '20 at 02:41

asked Apr 12 '20 at 02:20

Rudvick

I meant by this is that k is a composite number but couldn't find a notation for that. – Rudvick Apr 12 '20 at 02:22
Why don't you edit your question to include the text "$k$ is composite"? – Angina Seng Apr 12 '20 at 02:25
@AnginaSeng Thanks, Corrected it. – Rudvick Apr 12 '20 at 02:35
1

Earlier question by same user on similar sum: https://math.stackexchange.com/questions/3618219/efficient-computation-of-sum-i-1i-left-lfloor-sqrtn-right-rfloor – so, Rudvick, what is it with these sums involving floor functions? where are they coming from? – Gerry Myerson Apr 12 '20 at 03:55
@GerryMyerson Everyone is probably trying to solve this question(https://www.codechef.com/APRIL20A/problems/PPDIV) and it is a sub part of the problem. – Rudvick Apr 12 '20 at 06:46
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Thanks, Rudvick. What are the rules on codechef? Are people supposed to get help on the internet? – Gerry Myerson Apr 12 '20 at 06:49
@Rudvick: for what I have seen in codechef.com/APRIL20A/problems/PPDIV, I surmised that what is needed is a code to make the calculation, and obtain a concrete result, given "N". Is that it? if so, have a look of a code made in Mathematica 10 (with an example of what "N" is; here is taken as 30, but you can entry any positive intenger that you want; I changed "N" to "lastN", because "N" is a command in Mathematica 10): lastN = 30; Sum[ Floor[lastN/k] Boole[PrimeQ[k] == False], {k, 1, lastN}] – Arturo Ortiz Tapia Apr 13 '20 at 19:28

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