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APL (Dyalog Unicode), 30 29 bytes Saved 1 byte thanks to Razetime (could've saved 1 more with a tradfn, but I didn't feel like it) {∧/(0=1|+/÷≢)¨1↓¨g¨(g←∪⊢∨⍳)⍵} Try it online! This answer w...

posted 3y ago by user‭ · edited 3y ago by user‭

Answer

#4: Post edited by

user‭ · 2021-04-22T15:55:15Z (about 3 years ago)

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~~# [APL (Dyalog Unicode)], 30 bytes~~
~~{∧/(0=1|+/÷≢)¨1g¨0(g←∪⊢∨↓∘⍳)⍵}~~
~~[Try it online!][TIO-knrnm1lu]~~
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnm1lu]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OAZPWjjuX6Gga2hjXa@oe3P@pcpHlohWH6oRUGGulA2Ucdqx51LXrUseJR2@RHHTMe9W7WfNS7tfb//zSgKgUjBVMFMwVLBSNTBSMDcy6gmLEBkGUGxBYA "APL (Dyalog Unicode) – Try It Online"
This answer was incorrect before because it only checked if the number's *proper* divisors were cool, but it should work now.
Requires zero-indexing.
Explanation (to be updated):
```
{∧/(0=1|+/÷≢)¨1g¨0(g←∪⊢∨↓∘⍳)⍵}
(g←∪⊢∨↓∘⍳) ⍝ Define g to find divisors
⍳ ⍝ Make a range [0,n)
↓∘ ⍝ Drop the amount given on the left
⍝ Dropping 1 results in proper divisors,
⍝ dropping 0 results in all divisors
⊢∨ ⍝ GCD(n, x) for all x's in the range,
⍝ leaving us with divisors and a bunch of 1s
∪ ⍝ Remove duplicates
0 ⍵ ⍝ Apply this to ⍵ to get all divisors
¨ ⍝ For each of these divisors
1g ⍝ Find the proper divisors
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
```
~~### With trains, also 30 bytes~~
(∧/(0=1|+/÷≢)¨)1g¨0(g←∪⊢∨↓∘⍳)⊢
[Try it online!][TIO-knrnofx0]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnofx0]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OApMajjuX6Gga2hjXa@oe3P@pcpHlohaZh@qEVBhrpQOlHHasedS161LHiUdvkRx0zHvVu1gTy//9PO7TCUMFIwVTBTMFSwchUwcjAnAsoZmwAZJkBsQUA "APL (Dyalog Unicode) – Try It Online"

# [APL (Dyalog Unicode)], <s>30</s> 29 bytes
Saved 1 byte thanks to Razetime (could've saved 1 more with a tradfn, but I didn't feel like it)
{∧/(0=1|+/÷≢)¨1↓¨g¨(g←∪⊢∨⍳)⍵}
[Try it online!][TIO-knt2e200]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knt2e200]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OAZPWjjuX6Gga2hjXa@oe3P@pcpHloheGjtsmHVqQfWqGRDlTxqGPVo65FjzpWPOrdrPmod2vt//9pQDUKRgqmCmYKlgpGpgpGBuZcQDFjAyDLDIgtAA "APL (Dyalog Unicode) – Try It Online"
This answer was incorrect before because it only checked if the number's *proper* divisors were cool, but it should work now.
Requires zero-indexing.
Explanation (to be updated):
```
{∧/(0=1|+/÷≢)¨1g¨0(g←∪⊢∨↓∘⍳)⍵}
(g←∪⊢∨↓∘⍳) ⍝ Define g to find divisors
⍳ ⍝ Make a range [0,n)
↓∘ ⍝ Drop the amount given on the left
⍝ Dropping 1 results in proper divisors,
⍝ dropping 0 results in all divisors
⊢∨ ⍝ GCD(n, x) for all x's in the range,
⍝ leaving us with divisors and a bunch of 1s
∪ ⍝ Remove duplicates
0 ⍵ ⍝ Apply this to ⍵ to get all divisors
¨ ⍝ For each of these divisors
1g ⍝ Find the proper divisors
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
```
### With trains, 30 bytes
(∧/(0=1|+/÷≢)¨)1g¨0(g←∪⊢∨↓∘⍳)⊢
[Try it online!][TIO-knrnofx0]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnofx0]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OApMajjuX6Gga2hjXa@oe3P@pcpHlohaZh@qEVBhrpQOlHHasedS161LHiUdvkRx0zHvVu1gTy//9PO7TCUMFIwVTBTMFSwchUwcjAnAsoZmwAZJkBsQUA "APL (Dyalog Unicode) – Try It Online"

#3: Post edited by

user‭ · 2021-04-21T16:14:51Z (about 3 years ago)

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~~# [APL (Dyalog Unicode)], <s>30</s> 31 bytes~~
~~{∧/(0=1|+/÷≢)¨1g¨↑0(g←∪⊢∨↓∘⍳)⍵}~~
~~[Try it online!][TIO-knrnab25]~~
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnab25]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OAZPWjjuX6Gga2hjXa@oe3P@pcpHlohWH6oRWP2iYaaKQDFTzqWPWoa9GjDqDI5EcdMx71btZ81Lu19v//NKBCBSMFUwUzBUsFI1MFIwNzLqCYsQGQZQbEFgA "APL (Dyalog Unicode) – Try It Online"
This answer was incorrect before because it only checked if the number's *proper* divisors were cool, but it should work now.
Requires zero-indexing.
Explanation (to be updated):
```
(∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2
(∪⊢∨1,1↓⍳) ⍝ Find proper divisors
~~⍳ ⍝ Make a range [0,n)~~
~~1↓ ⍝ Drop the 0~~
~~⊢∨ ⍝ GCD(n, x) for all x's in the range,~~
~~⍝ leaving us with divisors and a bunch of 1s~~
~~∪ ⍝ Remove duplicates~~
⍣2 ⍝ Do this twice to get divisors of divisors
~~∘↑ ⍝ Turns into a vector first because of boxing~~
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
~~```~~

# [APL (Dyalog Unicode)], 30 bytes
{∧/(0=1|+/÷≢)¨1g¨0(g←∪⊢∨↓∘⍳)⍵}
[Try it online!][TIO-knrnm1lu]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnm1lu]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OAZPWjjuX6Gga2hjXa@oe3P@pcpHlohWH6oRUGGulA2Ucdqx51LXrUseJR2@RHHTMe9W7WfNS7tfb//zSgKgUjBVMFMwVLBSNTBSMDcy6gmLEBkGUGxBYA "APL (Dyalog Unicode) – Try It Online"
This answer was incorrect before because it only checked if the number's *proper* divisors were cool, but it should work now.
Requires zero-indexing.
Explanation (to be updated):
```
{∧/(0=1|+/÷≢)¨1g¨0(g←∪⊢∨↓∘⍳)⍵}
(g←∪⊢∨↓∘⍳) ⍝ Define g to find divisors
⍳ ⍝ Make a range [0,n)
↓∘ ⍝ Drop the amount given on the left
⍝ Dropping 1 results in proper divisors,
⍝ dropping 0 results in all divisors
⊢∨ ⍝ GCD(n, x) for all x's in the range,
⍝ leaving us with divisors and a bunch of 1s
∪ ⍝ Remove duplicates
0 ⍵ ⍝ Apply this to ⍵ to get all divisors
¨ ⍝ For each of these divisors
1g ⍝ Find the proper divisors
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
```
### With trains, also 30 bytes

(∧/(0=1|+/÷≢)¨)1g¨0(g←∪⊢∨↓∘⍳)⊢
[Try it online!][TIO-knrnofx0]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnofx0]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OApMajjuX6Gga2hjXa@oe3P@pcpHlohaZh@qEVBhrpQOlHHasedS161LHiUdvkRx0zHvVu1gTy//9PO7TCUMFIwVTBTMFSwchUwcjAnAsoZmwAZJkBsQUA "APL (Dyalog Unicode) – Try It Online"

#2: Post edited by

user‭ · 2021-04-21T16:05:05Z (about 3 years ago)

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~~# [APL (Dyalog Unicode)], 30 bytes~~
(∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2
~~[Try it online!][TIO-knrlg77h]~~
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrlg77h]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OApMajjuX6Gga2hjXa@oe3P@pcpHlohSZQcNWjrkWPOlYY6hg@apv8qHczUPhRx4xHbRMf9S42@v8/7dAKQwUjBVMFMwVLBSNTBSMDMyA25wKKGxsAWRYA "APL (Dyalog Unicode) – Try It Online"
I'm not totally sure about the validity of this answer since the BQN answer has different results, but if it isn't right, I will correct/delete it.
Requires zero-indexing.
~~Explanation:~~
```
(∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2
(∪⊢∨1,1↓⍳) ⍝ Find proper divisors
⍳ ⍝ Make a range [0,n)
1↓ ⍝ Drop the 0
⊢∨ ⍝ GCD(n, x) for all x's in the range,
⍝ leaving us with divisors and a bunch of 1s
∪ ⍝ Remove duplicates
⍣2 ⍝ Do this twice to get divisors of divisors
∘↑ ⍝ Turns into a vector first because of boxing
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
```

# [APL (Dyalog Unicode)], <s>30</s> 31 bytes
{∧/(0=1|+/÷≢)¨1g¨↑0(g←∪⊢∨↓∘⍳)⍵}
[Try it online!][TIO-knrnab25]
[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrnab25]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OAZPWjjuX6Gga2hjXa@oe3P@pcpHlohWH6oRWP2iYaaKQDFTzqWPWoa9GjDqDI5EcdMx71btZ81Lu19v//NKBCBSMFUwUzBUsFI1MFIwNzLqCYsQGQZQbEFgA "APL (Dyalog Unicode) – Try It Online"
This answer was incorrect before because it only checked if the number's *proper* divisors were cool, but it should work now.
Requires zero-indexing.
Explanation (to be updated):
```
(∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2
(∪⊢∨1,1↓⍳) ⍝ Find proper divisors
⍳ ⍝ Make a range [0,n)
1↓ ⍝ Drop the 0
⊢∨ ⍝ GCD(n, x) for all x's in the range,
⍝ leaving us with divisors and a bunch of 1s
∪ ⍝ Remove duplicates
⍣2 ⍝ Do this twice to get divisors of divisors
∘↑ ⍝ Turns into a vector first because of boxing
(∧/(0=1|+/÷≢)¨) ⍝ Check if divisors of divisors meet criteria
¨ ⍝ For every divisor's list of divisors
+/÷≢ ⍝ Calculate mean:
+/ ⍝ Sum
÷ ⍝ Divided by
≢ ⍝ Count
1| ⍝ Mod 1
0= ⍝ Is that 0? (0<x<1 if not integral)
∧/ ⍝ Is this true for all lists of divisors?
```

#1: Initial revision by

user‭ · 2021-04-21T15:19:43Z (about 3 years ago)

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# [APL (Dyalog Unicode)], 30 bytes

<!-- language-all: lang-apl -->

    (∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2

[Try it online!][TIO-knrlg77h]

[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-knrlg77h]: https://tio.run/##SyzI0U2pTMzJT///qG@qp/@jtgkG/9OApMajjuX6Gga2hjXa@oe3P@pcpHlohSZQcNWjrkWPOlYY6hg@apv8qHczUPhRx4xHbRMf9S42@v8/7dAKQwUjBVMFMwVLBSNTBSMDMyA25wKKGxsAWRYA "APL (Dyalog Unicode) – Try It Online"

I'm not totally sure about the validity of this answer since the BQN answer has different results, but if it isn't right, I will correct/delete it.

Requires zero-indexing.

Explanation:
```
(∧/(0=1|+/÷≢)¨)(∪⊢∨1,1↓⍳)¨∘↑⍣2
                (∪⊢∨1,1↓⍳)      ⍝ Find proper divisors
                         ⍳       ⍝ Make a range [0,n)
                       1↓       ⍝ Drop the 0
                   ⊢∨          ⍝ GCD(n, x) for all x's in the range,
                                ⍝ leaving us with divisors and a bunch of 1s
                 ∪              ⍝ Remove duplicates
                             ⍣2 ⍝ Do this twice to get divisors of divisors
                            ∘↑  ⍝ Turns into a vector first because of boxing
(∧/(0=1|+/÷≢)¨)                ⍝ Check if divisors of divisors meet criteria
             ¨                  ⍝ For every divisor's list of divisors
         +/÷≢                  ⍝ Calculate mean:
         +/                    ⍝ Sum
           ÷                   ⍝ Divided by
            ≢                  ⍝ Count
       1|                      ⍝ Mod 1
     0=                        ⍝ Is that 0? (0<x<1 if not integral)
 ∧/                            ⍝ Is this true for all lists of divisors?
```

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