Zumkeller numbers are the set of numbers whose divisors can be partitioned into two disjoint sets that sum to the same value. Each sum must contain divisor values that are not in the other sum, and all of the divisors must be in one or the other. There are no restrictions on ''how'' the divisors are partitioned, only that the two partition sums are equal.


;E.G.

: '''6''' is a Zumkeller number; The divisors '''{1 2 3 6}''' can be partitioned into two groups '''{1 2 3}''' and '''{6}''' that both sum to 6.

: '''10''' is not a Zumkeller number; The divisors '''{1 2 5 10}''' can not be partitioned into two groups in any way that will both sum to the same value.

: '''12''' is a Zumkeller number; The divisors '''{1 2 3 4 6 12}''' can be partitioned into two groups '''{1 3 4 6}''' and '''{2 12}''' that both sum to 14.


Even Zumkeller numbers are common; odd Zumkeller numbers are much less so. For values below 10^6, there is ''at least'' one Zumkeller number in every 12 consecutive integers, and the vast majority of them are even. The odd Zumkeller numbers are very similar to the list from the task [[Abundant odd numbers]]; they are nearly the same except for the further restriction that the abundance ('''A(n) = sigma(n) - 2n'''), must be even: '''A(n) mod 2 == 0'''


;Task:
:* Write a routine (function, procedure, whatever) to find Zumkeller numbers.
:* Use the routine to find and display here, on this page, the first '''220 Zumkeller numbers'''.
:* Use the routine to find and display here, on this page, the first '''40 odd Zumkeller numbers'''.
:* Optional, stretch goal: Use the routine to find and display here, on this page, the first '''40 odd Zumkeller numbers that don't end with 5'''.


;See Also:

:* '''[[oeis:A083207|OEIS:A083207 - Zumkeller numbers]]''' to get an impression of different partitions '''[[oeis:A083206/a083206.txt|OEIS:A083206 Zumkeller partitions]]'''
:* '''[[oeis:A174865|OEIS:A174865 - Odd Zumkeller numbers]]'''


;Related Tasks:

:* '''[[Abundant odd numbers]]'''
:* '''[[Abundant, deficient and perfect number classifications]]'''
:* '''[[Proper divisors]]''' , '''[[Factors of an integer]]'''

