[seqfan] Re: Did the OEIS already have the answer to this Math.StackExchange question?

[Charles Greathouse]

For all numbers n > 41 not of the form m*4^k where m is 2, 6, or 14, n can
be written as a sum of four positive squares. So excluding those three
infinite families, n has at least floor(sqrt(n - 41)) representations as a
sum of five positive squares. A rough upper bound on the number of these
families that it could touch is floor(log_2 (n/32)) + floor(log_2 (n/24)) +
floor(log_2 (n/14)) for a total upper bound of floor(sqrt(n - 41)) -
floor(log_2 (n/32)) - floor(log_2 (n/24)) - floor(log_2 (n/14)).

But even this nearly trivial bound is enough to prove that there is no
number with exactly 188 representations as the sum of five squares, since
at n = 48441 it gives a lower bound of 189. (OK, technically, there could
be problems with rounding that mean you'd need to check slightly higher.
But even then 50217 will suffice.) In fact checking to a million you should
be able to guarantee that further terms have at least 951 representations.



Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Fri, May 30, 2014 at 4:51 PM, M. F. Hasler <oeis at hasler.fr> wrote:

> PS: Having computed A025429(0..10^4) and found
> 33, 60, 105, 90, 132, 177, 145, 201, 225, ... =
> A080673 = Largest numbers with exactly n representations as sum of
> five positive squares.
>
> it is intriguing to look at that graph which suggests some similar
> regularity and a lower bound for   A080673 that grows to infinity, but
> it appears that
>
> 188,259,304,308,372,394,483,497,594,....
> are (maybe?) numbers such that there is no n with that number of
> representations!
> I got these zeros using a few thousand values of A025429, but it seems
> that:
>
> There is no number <= 10^6 that is the sum of five positive squares in
> exactly 188 ways. [Donovan Johnson, Aug 15 2013]
>
> If significant bounds could be given for A025429, then this list of
> zeros could be computed without doubts.
>
> Maximilian
>
> On Fri, May 30, 2014 at 3:56 PM, M. F. Hasler <oeis at hasler.fr> wrote:
> > On Fri, May 30, 2014 at 2:40 PM, Alonso Del Arte wrote:
> >> What is the largest integer with only one representation as a sum of
> five
> >> nonzero squares?
> >> http://math.stackexchange.com/questions/811824/
> >> This immediately suggests two or three sequences, which might already
> all
> >> be in the OEIS.
> >
> >
> > Sequence A025429 yields the
> > Number of partitions of n into 5 nonzero squares.
> >
> > Your suggested sequences and a link to that stackexchange page could be
> > added there.
> >
> > Maximilian
>
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