# [seqfan] Re: Exactly two subsets of proper divisors of n sum to n.

Charles Greathouse charles.greathouse at case.edu
Thu Oct 29 14:55:28 CET 2009

```Please submit it!  It seems to be a member of a natural sequence of sequences:
Abundant numbers with exactly 0 subsets of proper divisors that sum to
n: 70, 836, 4030, 5830, 7192, 7912, 9272, ... (Weird numbers A006037)
Abundant numbers with exactly 1 subsets of proper divisors that sum to
n: 20,78,88,102,104,114,138,174,186,222, ...
Abundant numbers with exactly 2 subsets of proper divisors that sum to
n: 12,18,42,56,66,100,176,196,208,348, ...

None have any small odd members, apparently.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Oct 29, 2009 at 8:07 AM, zak seidov <zakseidov at yahoo.com> wrote:
> Dear seqfans,
>
> This is a draft of (possible) submission,
> pending your YEAS AND NAYS...
>
> %N
> Exactly two subsets of proper divisors of n sum to n.
> %C
> Or numbers n such that A065205(n)=2.
> There are exactly 327 such numbers <10^4:
> 12,18,42,...,,9928,9948,9966, all even.
> %e
> n=12: proper divisors of 12 =  {1,2,3,4,6} and 12=2+4+6=1+2+3+6\Q
> n=9966: proper divisors of 9966 =  {1,2,3,6,11,22,33,66,151,302,453,906,1661,3322,4983} and\Q
> 9966=1661+3322+4983=302+453+906+3322+4983.
> %C
> Are there odd such numbers?
> Any theory behind this?
>
> %Y
> Cf. A065205 Number of subsets of proper divisors of n that sum to n.\Q
> A064771 Let S(n)=set of divisors of n,excluding n;
> sequence gives n such that there is a unique subset of S(n) that sums to n.\Q
> A065235 Odd numbers in A064771.
>
> %E
> Thanks ,
> Zak
>
>
>
>
>
>
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>

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