# Really Cool Numbers

Define a *cool* number as a number whose proper divisors (all except for the number itself) have an integral mean. Define a *really cool* number as a number whose divisors (including itself) are all cool. (We explicitly define 1 to be both cool and really cool.) Given a positive integer, determine whether or not it is really cool.

# Examples

Prime numbers are both cool and really cool, since 1 is defined as cool. 15 is really cool, because $\frac{1+3+5}{3} = 3$ and primes/1 are cool. 30 is cool, since $\frac{1+2+3+5+6+10+15}{7} = 6$, but not really cool, since 10 is not cool.

Here is a short list of really cool numbers for testing: $2, 5, 6, 9, 25, 207$

This is code-golf, so shortest code wins.

BQN, 25 bytesSBCS ``` ⌊⊸≡( …

~2mo ago

[Husk], 9 bytes ΛöS=⌊Ah …

~2mo ago

[APL (Dyalog Unicode)], 30 29 …

~2mo ago

## 3 answers

#
BQN, 25 bytes^{SBCS}

```
⌊⊸≡(/0=↕⊸|){(+´÷≠)∘𝔽¨«⟜𝔽}
```

This expression has a complicated structure. This link uses BQN's explain feature to show the order in which everything is applied. It's split into two expressions, where the `{`

on the left indicates to apply the modifier on the right.

```
⌊⊸≡(/0=↕⊸|){(+´÷≠)∘𝔽¨«⟜𝔽}
(/0=↕⊸|){ } # Operand 𝔽 to block modifier: proper divisors
↕ # Range 0,…,n-1
⊸| # before modular division
0= # equals zero
/ # Indices of ones
𝔽 # Apply the operand
«⟜ # Shift in the number itself
¨ # On each divisor:
∘𝔽 # Apply the operand again, then
(+´÷≠) # Mean (sum divided by length)
⌊⊸≡ # Floor matches argument
```

Much of the structure is composed of Before and After (`⊸⟜`

) and trains, with one block modifier. Note that Modulus (`|`

) has its arguments reversed relative to the modular division operator `%`

in many languages: `3|5`

is 2, for example. The function `/0=↕⊸|`

gives proper divisors including 1, but for testing which divisors are cool we want to include the number itself and exclude 1 (we know it's cool). Shifting in the original number on the right side accomplishes this.

#### 2 comments

@user But 206 itself isn’t cool, since 106 isn’t divisible by 3

@Quintec Oh, I just realized that the really cool definition includes all divisors, not just the proper divisors.

#
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←∪⊢∨⍳)⍵}
```

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←∪⊢∨↓∘⍳)⊢
```

#### 1 comment

-1 simplifying the inner function. You can also get -2 by converting to a tradfn, since ⍵ is used only once.

# Husk, 9 bytes

```
ΛöS=⌊AhḊḊ
```

Try it online! or Verify all testcases

returns number of divisors + 1 for true and 0 for false.

#### 2 comments

First link goes to the APL solution.

whoops sorry bout that

## 0 comments