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 ``` ⌊⊸≡( …
3y ago
[Husk], 9 bytes ΛöS=⌊Ah …
3y ago
[APL (Dyalog Unicode)], 30 29 …
3y ago
3 answers
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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←∪⊢∨⍳)⍵}
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←∪⊢∨↓∘⍳)⊢
BQN, 25 bytesSBCS
⌊⊸≡(/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.
Husk, 9 bytes
ΛöS=⌊AhḊḊ
Try it online! or Verify all testcases
returns number of divisors + 1 for true and 0 for false.
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