[seqfan] Re: 448 ways

Allan Wechsler acwacw at gmail.com
Sun Nov 19 17:33:19 CET 2017


At least in https://oeis.org/A295159, the sequence we started with, the
word "nonnegative" could be removed, because all integer squares are
nonnegative. But it should not be changed to "nonzero". For example
A295159(1) = 0; that is, the smallest number that is the sum of five
nonnegative squares in exactly one way is 0, because 0 = 0+0+0+0+0. But 0
is not the sum of five nonzero squares in any way whatsoever. Similarly,
A295159(3) = 13, because 13 = 9 + 4 + 0 + 0 + 0 = 9 + 1 + 1 + 1 + 1 = 4 + 4
+ 4 + 1 + 0.  If the word "nonnegative" in the title were changed to
"nonzero", two of these representations would be invalid.

A295159(n) is the smallest k such that A000174(k) = n. If we replace
"nonnegative" with "nonzero", we would need to replace A000174 with
https://oeis.org/A025429.



On Sun, Nov 19, 2017 at 10:36 AM, Neil Sloane <njasloane at gmail.com> wrote:

> Could the explanation (why on earth would someone say "nonnegative
> square"?)
> be that they meant to say "nonzero sequence"?  It's not a typo, it's a
> braino.    So the fix would be to replace
> "nonnegative square" with "nonzero square" ??
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Sat, Nov 18, 2017 at 12:10 PM, Peter Munn <techsubs at pearceneptune.co.uk
> >
> wrote:
>
> > Dear seqfans,
> >
> > On Thu, November 16, 2017 6:09 pm, Allan Wechsler wrote:
> > > Very interesting! So we need a new sequence, "numbers N for which no K
> > is
> > > the sum of five squares in exactly N ways", of which 448 will be the
> > leading entry.
> >
> > Another sequence in which 448 would be the first term is "least m for
> > which no K is the sum of n squares in exactly m ways", which would have
> an
> > offset for n of 5.  Maybe it would be easier to calculate sufficient
> terms
> > for this?
> >
> > By the way, how do others see the "nonnegative" qualification for
> "square"
> > in the name of A295159?
> >
> > 19 sequences showed up in the OEIS search as having "nonnegative square"
> > in their name (when excluding those also having "root" in the name), but
> > only 7 of them date from before this month.
> >
> > Four of these, A229062, A073613, A160053 and A180161, do so with clear
> > reference to A001481, which no longer has the qualification in its own
> > name, "Numbers that are the sum of 2 squares."  Another, A230314,
> "Numbers
> > that are simultaneously the sum of two nonnegative squares and the sum of
> > two nonnegative cubes", benefits by pointing out a symmetry.
> >
> > Of the remaining two, A006431, "Numbers that have a unique partition into
> > a sum of four nonnegative squares" is clearly the one related to A295159.
> > The other, A166498, has had its "nonnegative" deleted following my
> > questioning it when making a correction.
> >
> > Best Regards,
> >  Peter Munn
> > > On Thu, Nov 16, 2017 at 7:14 AM, Giovanni Resta
> > > <giovanni.resta at iit.cnr.it>
> > > wrote:
> > >> On 11/16/2017 03:51 AM, israel at math.ubc.ca wrote:
> > >> To make that lower bound 448 would require x >= 2339578, and that's
> too
> > >>> big for me to search. But maybe better lower bounds are known.
> > >> I checked all the x between 25000 and 2339600 and all have at least
> 449
> > decompositions as a sum of five squares.
> > >> Giovanni
> > >> --
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> >
> >
> >
> >
> >
> >
> >
> > --
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> >
>
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