Erdős-Nicolas numbers

Erdős-Nicolas numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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 greater than one.

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

• OEIS:A194472 - Erdős–Nicolas numbers

J

Implementation:<lang J>divisors=: {{ /:~ ,*/@> { (^ i.@>:)&.>/__ q: y}} ::_: erdosnicolas=: {{ y e. +/\ _2}. divisors y }}"0</lang>Task example:<lang J> I.erdosnicolas i.1e7 24 2016 8190 42336 45864 392448 714240 1571328

  (,. 1++/\@divisors i. ])@>24 2016 8190 42336 45864 392448 714240 1571328
    24   6
  2016  31
  8190  43
 42336  66
 45864  66
392448  68
714240 113

1571328 115</lang>

Phix

Translation of: Wren

with javascript_semantics
function erdos_nicolas(integer n)
    -- (The default for factors() is no 1 nor n,
    --  though you can change that if you want,
    --  ie ",1" -> 1 and n; ",-1" -> 1 but not n.)	
    sequence divisors = factors(n)
    integer tot = 1
    for i=1 to length(divisors)-1 do
        tot += divisors[i]
        if tot=n then return i+1 end if
        if tot>n then exit end if
    end for
    return 0
end function
 
constant limit = 8
integer n = 2, count = 0
while count<limit do
    integer k = erdos_nicolas(n)
    if k>0 then
        printf(1,"%d from %d\n",{n,k})
        count += 1
    end if
    n += 2
end while

Output same as Wren

Wren

Library: Wren-math

<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 (true) {

   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