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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

How can i find the partition for a zumkeller number? Is there a theorem to help find the partition and sum of zumkeller numbers?

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,281

hi NakulG

I don't know the answer to this but have a look here:

Creating code to 'number crunch' the sequence suggests that there is no algorithm for this.

Bob

EDIT: If you have the set of factors, f subscript i, then the sum will be (∑f)/2. So you can dismiss some possibles straight away as n will always be a factor and hence ∑f must be ≥ n

eg. 12: {1,2,3,4,6,12} ∑f = 28 so sum = 14. Try 12 + 2 as one partition. Notice that 1+3+4+6 = 14 as well.

But eg. 15: {1,3,5,15} ∑f = 24 so sum = 12. This cannot be a z number as 15 > 12.

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi NakulG;

The best computational way is to use a generating function.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Apologies for my ignorance, what is a generating function

*Last edited by NakulG (2015-04-03 01:37:00)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

A generating function is simply a clothesline that we hang the coefficients of our polynomial on. It is a power series where issues of convergence do not arise. Although discovered by De Moivre it was used by Euler for partition problems.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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