Numbers that are sums of (distinct) squares in exactly k ways

[David W. Wilson]

At first glance, it looks as if 2n^2 is in A025305 but not in A025287 for n
in A084648.

Also, 2^5^(2k) has k distinct and k+1 nondistinct representations, so

Numbers that are the sum of 2 nonzero squares in exactly k ways

and

Numbers that are the sum of 2 distinct nonzero squares in exactly k ways

will differ for any k >= 0.

> -----Original Message-----
> From: Ray Chandler [mailto:rayjchandler at sbcglobal.net]
> Sent: Wednesday, May 28, 2008 7:39 PM
> To: seqfan at ext.jussieu.fr
> Cc: 'Richard Mathar'
> Subject: RE: Numbers that are sums of (distinct) squares in exactly k
> ways
> 
> 8450 is the first term in A025305 but not A025287.
> 
> 31250 is the first term in A025287 but not A025305.
> 
> Ray Chandler
> 
> > -----Original Message-----
> > From: Richard Mathar [mailto:mathar at strw.leidenuniv.nl]
> > Sent: Wednesday, May 28, 2008 2:08 PM
> > To: seqfan at ext.jussieu.fr
> > Subject: Numbers that are sums of (distinct) squares in exactly k
> ways
> >
> >
> > We have some sequences of numbers that can be written as a sum
> > of squares in in exactly k (k variable) ways, with or without
> > the restriction that these squares are distinct, for example
> >
> > http://research.att.com/~njas/sequences/?q=id:A25305|id:A25287
> >
> > Q: when do these start to differ? Is there a reason of symmetry why
> > a number n, which is a sum of nonzero squares in exactly 4 ways,
> > cannot have one representation amongst these where n=2*x^2?
> >
> >
> > The challenge is either to proof that these sequences are the same,
> > or to find some least number which is in one sequence but not
> > the other.
> >
> > Richard Mathar http://www.strw.leidenuniv.nl/~mathar
>