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## #1 2015-04-01 04:41:30

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

### How can i find the partition for a zumkeller number?

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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## #2 2015-04-01 05:48:53

bob bundy
Registered: 2010-06-20
Posts: 8,170

### Re: How can i find the partition for a zumkeller number?

hi NakulG

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

https://oeis.org/A083207

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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## #3 2015-04-01 12:13:42

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

### Re: How can i find the partition for a zumkeller number?

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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## #4 2015-04-03 01:35:45

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

### Re: How can i find the partition for a zumkeller number?

Apologies for my ignorance, what is a generating function

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

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## #5 2015-04-03 08:54:05

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

### Re: How can i find the partition for a zumkeller number?

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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