Erdős-Nicolas numbers
- Definition
An Erdős–Nicolas number is a positive integer which is not perfect but is equal to the sum of its first k divisors (arranged in ascending order and including one) for some value of k.
- Examples
24 is an Erdős–Nicolas number because the sum of its first 6 divisors (1, 2, 3, 4, 6 and 8) is equal to 24 and it is not perfect because 12 is also a divisor.
6 is not an Erdős–Nicolas number because it is perfect (1 + 2 + 3 = 6).
48 is not an Erdős–Nicolas number because its divisors are: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The first seven of these add up to 36, but the first eight add up to 52 which is more than 48.
- Task
Find and show here the first 8 Erdős–Nicolas numbers and the number of divisors needed (i.e. the value of 'k') to satisfy the definition.
- Stretch
Do the same for any further Erdős–Nicolas numbers which you have the patience for.
- Note
As all known Erdős–Nicolas numbers are even you may assume this to be generally true in order to quicken up the search. However, it is not obvious (to me at least) why this should necessarily be the case.
- Reference
Wren
<lang ecmascript>import "./math" for Int
var erdosNicolas = Fn.new { |n|
var divisors = Int.properDivisors(n) // excludes n itself
var dc = divisors.count
if (dc < 3) return 0
var sum = divisors[0] + divisors[1]
for (i in 2...dc-1) {
sum = sum + divisors[i]
if (sum == n) return i + 1
if (sum > n) break
}
return 0
}
var limit = 8 var n = 2 var count = 0 while (count < limit) {
var k = erdosNicolas.call(n)
if (k > 0) {
System.print("%(n) from %(k)")
count = count + 1
if (count == limit) return
}
n = n + 2
}</lang>
- Output:
24 from 6 2016 from 31 8190 from 43 42336 from 66 45864 from 66 392448 from 68 714240 from 113 1571328 from 115