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Proper Divisor mash-up

A proper divisor is a divisor of a number n, which is not n itself. For example, the proper divisors of 12 are 1, 2, 3, 4 and 6.

You will be given an integer x, x ≥ 2, x ≤ 1000. Your task is to sum all the highest proper divisors of the integers from 2 to x (inclusive) (OEIS A280050).

Example (with `x = 6`):

Find all the integers between 2 and 6 (inclusive): 2,3,4,5,6.
Get the proper divisors of all of them, and pick the highest ones from each number:
- 2 -> 1
- 3 -> 1
- 4 -> 1, 2
- 5 -> 1
- 6 -> 1, 2, 3.
Sum the highest proper divisors: 1 +たす 1 +たす 2 +たす 1 +たす 3 =わ 8.
The final result is 8.

Test Cases

Input | Output
-------+---------
 |
 2 | 1
 4 | 4
 6 | 8
 8 | 13
 15 | 41
 37 | 229
 100 | 1690
 1000 | 165279

Rules

Default Loopholes apply.
You can take input and provide output by any standard method.
This is code-golf, the shortest valid submission in each language wins! Have Fun!

Answer*

# [Python 2], 50 bytes



 f=lambda n,k=2:n/k and(f(n,k+1),n/k+f(n-1))[n%k<1]

This is *slow* and can't even cope with input **15** on TIO.

[Try it online!][TIO-j4besawb]

However, memoization (*thanks @musicman523*) can be used to verify all test cases.

[Try it online!][TIO-j4c4ucac]

### Alternate version, 52 bytes

At the cost of 2 bytes, we can choose whether to compute `f(n,k+1)` or `n/k+f(n-1)`.

 f=lambda n,k=2:n>1and(n%k and f(n,k+1)or n/k+f(n-1))

With some trickery, this works for all test cases, even on TIO.

[Try it online!][TIO-j4begnqc]

[Python 2]: https://docs.python.org/2/
[TIO-j4besawb]: https://tio.run/##DYxBCoMwFETX9RSzCSYY2yZIKVJ7kdKFrUqD@iNJXHj69A/M4j2Y2Y7082SzWzcfEuIRC@45jimM3z1E52lxq0vSXDkqT93Sr5@hB@m5sy1dZvQ0yEkyV0ZpFhVDbZR6kZgf5p0nH0BwBKvRaNw07m1x2oKjJEthdtRPCLuXEOAX8Foplf8 "Python 2 – Try It Online"
[TIO-j4c4ucac]: https://tio.run/##VU5Ra8IwGHw2v@Jeis2MrK3ODamDMRiUbT5svgwRcTbBUJuUNH1wY7@9pikie8mXu/vu7qtO9qBV0gqjSxhe68bsOWRZaWNxQzxdn@orQ2puzVGW0oYfb9l7ttp@rp6eXxk83GbLl2yZrb4Y/kFKvY/vG1NLrXp/jDTFJKKtWBx35Xe@g2LFIpmr2wI7lYcidHgUU@aIkQPjmNK1Coo03rQ5Fyh5qeUPDwWdk0EHsMDvHxl02oEfK25cAArHJt3GQAp4gkJpC6l8QCd487rXNm5d9F@nGG4b0y9edHIh@wpChHNcT3FYG6guPmGYMswYHhjiO4bJvZtR5J/I9VZGKhsOg2mD8SOCWTNE4C90/ZTS9gw "Python 2 – Try It Online"
[TIO-j4begnqc]: https://tio.run/##VYxNi8IwGITP9lfMpdhoxMb6haggCwth1YP24kmqthhq35QkPfjr3dg9LF5meB6YqZ/uril5FUZXMLnVjbnmUFWtjUMvaLV92n8T2NyZh6qUiw5buZPp@Zhuvn44WjzL/bfcy/TE8YGMtbv82hirNP3tBZZLJDF7FatHVl1uGYiXq9GC1iKjW0RhCd8oIq/7gmkDGpZ9jwPB/OjNUIQRx5hjyjHnEBOOZOY7jtuIF0GnNopc1A3HDQZrhNOmixD@833M/NEv "Python 3 – Try It Online"

If this is an answer to a challenge…

…Be sure to follow the challenge specification. However, please refrain from exploiting obvious loopholes. Answers abusing any of the standard loopholes are considered invalid. If you think a specification is unclear or underspecified, comment on the question instead.
…Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one. Explanations of your answer make it more interesting to read and are very much encouraged.
…Include a short header which indicates the language(s) of your code and its score, as defined by the challenge.

More generally…

…Please make sure to answer the question and provide sufficient detail.
…Avoid asking for help, clarification or responding to other answers (use comments instead).

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\$\begingroup\$ Since f is a pure function, you can memoize it to run the larger cases on TIO \$\endgroup\$

musicman523
– musicman523

2017年06月25日 02:55:02 +00:00
Commented Jun 25, 2017 at 2:55
\$\begingroup\$ Right, not being able to use a decorator threw me off. Thanks! \$\endgroup\$

Dennis
– Dennis

2017年06月25日 03:06:58 +00:00
Commented Jun 25, 2017 at 3:06

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Proper Divisor mash-up

Example (with x = 6):

Test Cases

Rules

Answer*

Example (with `x = 6`):